This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 27: Principles embedded in this site
A physical understanding is completely unmathematical, imprecise, an inexact thing but absolutely necessary to a physicist. Richard Feynman: Lectures on Physics II Chapter 2: Differential Calculus of Vector Fields
Research is to see what everybody has seen and think what nobody has thought. Peter Osper (1957): Review: Albert Szent-Györgyi: Bioenergetics
Perhaps now is the time to widen the quest for understanding still further, to expand the intellectual effort beyond conventional science—to engage the larger non-scientific communities of philosophers, theologians and others who often resonate with the cosmic-evolutionary theme even if not in name, all in an ambitious effort to construct a millennial world view of who we are, where we came from, and how we fit into the cosmic scheme of things as wise, ethical, human beings. Eric J. Chaisson (2002): Cosmic Evolution: The Rise of Complexity in Nature, page 211
Contents
Principle 1: The initial singularity: Nothing comes from nothing
Principle 2: Zero sum complexification
Principle 3: The complexity of the Universe is generated through evolution by variation and selection
Principle 4: Each step in evolution requires a dynamic base with kinematic variation
Principle 5: Random evolutionary process may discover NP functions to be copied by P processes
Principle 6: A truly divine Universe is constrained only by the principle of non contradiction
Principle 7: Symmetry with respect to complexity
Principle 8: The heuristic of simplicity
Principle 9: Gravitation is code free universal communication
Principle 10: Addressable communication requires one to one contact
Principle 11: Formal consistency is selected by quantum mechanics
Principle 12: Creation: Gravitation provides energy to convert kinematic formalism into dynamic reality
Principle 13. The information carried by a physical state is equal to the entropy of the space it occupies
Principle 14: Quantization is a certainty principle, not an uncertainty principle
Principle 15: An unmodulated continuum carries no information
Principle 16: Requisite variety: the measure of information is the measure of control
Principle 17: Theology is the comprehensive theory of everything derived from physical data
Principle 1: The initial singularity: nothing comes from nothing
Where did we come from? Our ancestors. And where did they come from? . . . There are just two answers. The ancestors have existed forever, or they were created some time in the past. If they were created, where did the Creator come from? Ultimately, if we stand by the proposition that nothing comes from nothing, we are led to propose that either the world or its creator is eternal, that is, had no beginning (and quite possibly, no end).
The ancient foundations of Christian theology are to be found in the Hebrew Bible. The first book, Genesis, opens with the words In the beginning God created the heavens and the Earth. Aquinas, with the help of Aristotle concluded that God is an absolutely simple structureless entity of pure action existing outside space and time. Hebrew Bible - Wikipedia, Thomas Aquinas, Summa, I, 2, 3: Does God exist?
Modern physics has come to a similar conclusion. Penrose, Hawking and Ellis studied Einstein's classical general theory of relativity and found that it predicts the existence of structureless singularities, points outside the universe of space and time. Some imagine a "big bang" which marked the explosive emergence of the Universe from an initial singularity. Hawking & Ellis (1975): The Large Scale Structure of Space-Time, Big Bang - Wikipedia
The initial singularity derived from the classical theory of relativity is a classical object. Attempts to develop a quantum theory of relativity, and a corresponding a quantum initial singularity have so far failed.
Here I assume that a quantum initial singularity would be identical to the traditional model of God developed by Aquinas. This God and the singularity share four properties: they are eternal; they are pure action; they are without structure, outside space and time; and they are the omnipotent source of the Universe.
The development of this site proceeds on this hypothesis. It exploits some ideas from quantum theory to understand how the Universe we experience might have developed inside a divine initial singularity under the influence of unbounded action and local consistency.
Principle 2: Zero sum complexification
The eternal initial singularity which creates the Universe within itself starts with no knowledge and an empty mind and retains this property forever. We identify its initial state, identical to the traditional Christian God, with pure gravitation, which we identify as an empty set which under the influence of fixed point theory develops Hilbert space and quantum mechanics. Under the influence of the kinematic quantum mechanical development of consistent eigenstates, the primordial gravitation bifurcates into potential and kinetic energy, the kinetic energy converts the kinematic eigenstates into real dynamic particles which come into existence, by analogy to the persons of the Trinity, independently of the initial singularity. (see page 17: Gravitation and quantum theory—in the beginning)
The continued complexification of the Universe proceeds in similar fashion. The next zero sum bifurcation may be the production two classes of particles, fermions and bosons, leaving the symmetry of potential and kinetic energy unchanged since both bosons and fermions carry energy. The next step in the complexification of the Universe may be the emergence of Minkowski space. (See page 12: The quantum creation of Minkowski space
This concept is highly speculative, but is motivated by analogy with the notion that the total energy of the Universe remains at all times zero, suggesting that, consistent with its divine nature, it remains forever one and simple.
Principle 3: The complexity of the Universe is generated through evolution by variation and selection
In scientific investigations . . . it is permitted to invent any hypothesis, and if it explains various large and independent classes of facts, it rises to the rank of a well grounded theory. Charles Darwin (1875): The Variation of Animals and Plants Under Domestication
How did the enormously complex Universe come to be from the structureless initial singularity? Here I propose a process of evolution driven by the omnipotence of the initial singularity. We say an entity is omnipotent if it can do anything that does not involve contradiction. Evolution proceeds by random variation and deterministic selection. Entities that are not eternal survive for extended periods if they are capable of reproducing themselves. Aquinas, Summa I, 25, 3: Is God omnipotent?
I assume that it is generally agreed (at least in scientific circles) that evolution is adequate to explain the billions of organisms that have inhabited the tree of life. Many of these organisms comprise trillions of cells which are precise function molecular arrangements of millions or billions of atoms.
Given the existence of life, the evolutionary hypothesis of Darwin and Wallace is capable of explaining the origin of new species. Here I propose that given an omnipotent initial singularity analogous to the traditional living God, an evolutionary mechanism exploiting the information processing power of quantum theory is capable of explaining the emergence of the Universe within the initial singularity.
Many steps along the way are described in the previous 25 pages of this site. In particular the promise of an evolutionary approach to heaven on Earth by analogy to the political structure of our own bodies described on Page 25: Conclusion—political constitution built on human symmetry.
Principle 4: Each step in evolution requires a dynamic base with kinematic variation
Evolution among living creatures depends on their duality: a kinematic genome contained in a dynamic living body capable of decoding the genome in order to construct itself. We imagine that this structure exists at all stages in the evolution of the universe, beginning with the dynamic living initial singularity within which Hilbert space and quantum mechanics work to identify stationary states which can be realized with energy drawn from gravitational potential as described on page 17: Gravitation and quantum theory—in the beginning.
Principle 5: Random evolutionary process may discover NP functions to be copied by P processes
The creative random variation of the evolving Universe is not constrained to computable functions but may discover NP functions. The reliable reproduction of these structures, however, requires the testing and reproduction of these discoveries by computable functions of the P type level of complexity. P versus NP problem - Wikipedia
Principle 6: A truly divine Universe is constrained only by the principle of non contradiction
The divinity is traditionally omnipotent, and to be omnipotent is to be constrained only by the inability to create an actual contradiction. Aquinas, Summa I, 25, 3: Is God omnipotent?
Principle 7: Symmetry with respect to complexity
We learn from the Christian doctrine of the Trinity that Gods produce Gods within themselves: page 8: The theology of the Trinity. Christianity limits this process to three. We place no limit. From this we derive the key principle of this site.
Its historical origin lies in Werner Heisenberg's visit to Helgoland while he was studying with Bohr. There he first conceived the idea of ignoring the electronic mechanism of the atom proposed by Bohr and concentrating on just the observables, the frequencies and line weights of the measured atomic spectrum. His idea was developed with Born and Jordan into matrix mechanics and led to the expansion of the concept of observable from single real number in classical physics to a set (possibly infinite) of numbers corresponding to the eigenvalues of a matrix. These values could, for instance, refer to all the individual lines in an atomic spectrum.
Born realized that the relative frequency of observation of particular lines determined their weight, which depended in turn on the distance (inner product) in Hilbert space between their eigenvalues. This concept determines the frequency of observation of particular events and is now known as the Born Rule. Born rule - Wikipedia
From the communication theoretical point of view, a quantum observable is an information source A with an alphabet of i symbols (eg spectral lines) ai whose probabilities pi must add up to 1. This ensures that from a probabilistic point of view the observations of the symbols constitute a complete or collectively exhaustive system of events. Collectively exhaustive events - Wikipedia
This requirement is a fundamental constraint on mathematical treatments of quantum systems. Von Neumann uses it to prove the equivalence of the wave and matrix representations of quantum mechanics. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics chapter 1.4: The equivalence of the two systems - Hilbert space
In theory this constraint applies not only to the outcome of quantum events, but to all communications, no matter how complex, including conversations between people. It is, in other words, a symmetry with respect to complexity. In quantum mechanics the probabilities of outcomes are normalized by the unitarity of computations in the linear algebra of Hilbert space. We might conjecture that the same mathematical structures hold at all scales insofar as quantum mechanics in Hilbert space is the formal mechanism underlying all observable processes in the Universe. (see also page 24: The end of fields theory §8: Symmetry with respect to complexity). Unitarity (physics) - Wikipedia
In sum, the interactions between particles in quantum mechanics are formally equivalent to conversations between people and to any other communications described by Shannon's mathematical theory. One conclusion of the theory of communication is that error in communications through noisy channels can be reduced to zero at the cost of quantization and a corresponding reduction in speed: see page 11: Quantization: the mathematical theory of communication. Claude Shannon (1949): Communication in the Presence of Noise
Principle 8: The heuristic of simplicity
It is often said that God is perfect, beyond our understanding. We cannot say what God is, only what they are not. This approach to God is called apophatic, from the Greek apophemi to deny. Apophatic theology - Wikipedia
The apophatic approach relates to another traditional feature of the Christian God, its absolute simplicity. From a modern point of view information is represented by physical marks like these letters. On page 11: Quantization: the mathematical theory of communication we see that an absolutely simple God has no means of representing information and cannot therefore be omniscient. Because this God is absolutely simple there is really nothing to understand, so God is well within the ambit of human intelligence. This suggests that must reject Bernard Lonergan's idea (very common in Christian theology) that there is a category of transcendent knowledge beyond human understanding. Aquinas, Summa, I, 3, 7: Is God altogether simple?, Bernard Lonergan (1992): Insight: A Study of Human Understanding, pp. 657 sqq.
There is little to be said about a one state system other than that it exists. The next level of complexity is two state systems, which are relatively easy to understand and are, quantum mechanically, very rich in application. The heuristic of simplicity suggests that we should not ignore simple things because our Universe was, in the beginning, a very simple thing, as simple as God. Feynman lectures on physics: III:09: Chapter 9: The Ammonia Maser
Principle 9: Gravitation is code free universal communication
Philosophy is written in this grand book - the Universe, which stands continually open before our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics . . . Galileo Galilei (1610, 1957): Discoveries and Opinions of Galileo: Including the Starry Messenger, p 238.
Isaac Newton justified Galileo's choice of language. His mathematics worked perfectly but he hit a well known philosophical snag: action at a distance. Since Newton's time physicists have filled spacetime with various media: ether, field, vacuum, condensate and so on. All these are intended to establish causality between distant objects like the Sun and the planets.
The most reasonable and observable of these means of contact is gravitation. Einstein showed that gravitation is not something in space, it is a property of space itself. There is a widespread feeling in quantum field theory that gravitation should also be a field in space, mediated by undetectable particles called gravitons. This seems to be an unnecessary complication.
As Newton realized, gravitation is universal. It sees only energy in whatever form. It is an ubiquitous feature of spacetime and determines the large scale structure of the Universe. From the point of view of this site it seems reasonable to identify it with the initial singularity (ie our analogy of god) before the emergence of spacetime and matter. If this is case we always feel the divine presence as weight.
It may be imagined that conservation of energy requires that all the energy in the current universe is present in the initial singularity to power the big bang. This is hard to believe, since the existence of energy and momentum seems to be coupled to the existence of spacetime. The classical initial singularity precedes spacetime. An alternative view is that the total energy of the universe is zero. Richard Feynman (2002): Feynman Lectures on Gravitation, Zero-energy universe - Wikipedia
With the emergence of quantum theory gravitation bifurcates so that the potential energy carried by gravitation becomes exactly equal and opposite to the energy of the particles that constitute the visible universe (see page 16: Potential + kinetic = zero energy universe, and page 17: Gravitation and quantum theory—in the beginning). We guess that gravitation itself is structureless; all the structure in the Universe including the four dimensions of of spacetime arise from quantum theory. In his article on the field equations of gravitation Einstein draws attention to two points. First, the postulate of relativity in its most general formulation makes spacetime coordinates into physically meaningless parameters; and second that the postulate of general relativity cannot reveal to us anything new and different about the essence of various processes in nature than what the special theory of relativity taught us already. This supports the idea on page 12: The quantum creation of Minkowski space that the Minkowski metric is a quantum construction motivated by gravitation. Albert Einstein (1915): The Field Equations of Gravitation
Principle 10: Addressable communications require one to one contact
Because it is ubiquitous and structureless gravitation has no particular address. I cannot explain the quantum mechanical algorithm that establishes the Minkowski metric, but given the emergence of quantum mechanics it seems reasonable to assume that evolution, given the enormous variety available to it in Hilbert space, could have selected the vectors and operators necessary to create this metric. It seems that Minkowski space is a consequence of the quantum mechanical discovery of bosons and fermions. A defect I see in the big bang theory is that it seems to assume the existence of spacetime in the beginning. This is inconsistent with the idea that the initial singularity is structureless (and maybe still exists with us in the form of naked gravitation). Boson - Wikipedia, Fermion - Wikipedia
As Einstein points out in his paper on the special theory, Minkowski space is formally kinematic, but we can expect it to become dynamic with the emergence of massless and massive particles.
Once we have Minkowski spacetime we have null geodesics and massless bosons carrying signals between fermions. The Pauli exclusion principle seems to require 3 spatial dimensions to enable fermions to move freely without occupying the same point in spacetime. Since particles are discrete objects we may assume that communication and interaction in spacetime is managed by contact between individual particles without the necessity to postulate fields. Fermions communicate through massless bosons which follow null geodesics, carrying quantum states from one fermion to another in spacetime. We imagine that such contacts involving the creation and annihilation of bosons are one to one and involve the exchange of one quantum of action.
Here we can proceed by an analogy made possible by principle of symmetry with respect to complexity described in Principle 7. Let us assume that all particles in the Universe, running from elementary particles to ourselves and beyond, are images of the initial singularity, that is to say a dynamic shell enclosing a kinematic Hilbert space. On this analogy my observable body is my shell, my mind is my Hilbert space. It is in effect outside spacetime, the source of the variation which I explore to determine what I will do next.
As I communicate quantum mechanically with other people, my inner state is changed and we assume that the same is true for electrons. Since Hilbert space is prior to spacetime, an electron, even if we assume it has zero size, can still have an interior Hilbert space like myself or the initial singularity which can determine its behaviour.
The strength of electric field is the rate at which it creates and annihilates photons, a quantum computational process. We may therefore imagine that the electric field is not so much a spacetime phenomenon that varies as 1/r as something more like software that determines the rate of interaction with photons. This may explains the fact that the quantum interactions between electrons and photons are approximately 1040 times more frequent than the rate of interaction of electrons with gravitation.
Principle 11: Formal consistency is selected by quantum mechanics
The apparent presence of a final cause (in the Aristotelian sense) in evolution arises not because successful reproduction is designed into the genome, but because selection in a particular environment picks out those genotypes that actually survive. The answer proposed on this site to deal with the complexities introduced by trying to unify quantum mechanics and special relativity is that we consider Hilbert space to be a kinematic process driven by a dynamic system. This does not mean that the dynamic system controls the Hilbert space, it simply activates it. The roles of quantum mechanics is to select stationary states, ie eigenstates, from the possibilities provided by Hilbert space. Given their formal consistency, such states have the potential to exist, and in the picture presented here take energy from gravitational potential to become dynamic entities, ie particles.
Feynman's path integral representation of quantum mechanics is consistent with this idea. Every quantum event is precisely measured by a quantum of action, and the formalism of quantum theory requires that any vector in Hilbert space, no matter how many other vectors are added together to create it, is normalized by its inner product with itself to an absolute value of 1, this establishes its over all probability to 1. It is therefore a stationary state since an event with probability 1 is guaranteed to happen. This explains the quantum mechanical emergence of symmetries and conservation laws:
In classical physics there are a number of quantities which are conserved—such as momentum, energy, and angular momentum. Conservation theorems about corresponding quantities also exist in quantum mechanics. The most beautiful thing of quantum mechanics is that the conservation theorems can, in a sense, be derived from something else, whereas in classical mechanics they are practically the starting points of the laws. . . . In quantum mechanics, however, the conservation laws are very deeply related to the principle of superposition of amplitudes, and to the symmetry of physical systems under various changes. Feynman lectures on physics FLP III_17: Chapter 17: Symmetry and Conservation Laws
Principle 12: Creation: Gravitation provides energy to convert kinematic formalism into dynamic reality
We understand the initial singularity to be a real dynamic entity whose fundamental nature is most closely defined by Einstein's general theory of relativity. From a mathematical point of view we approximate this as a set that fulfills the conditions of Brouwer's fixed point theorem. We understand the action corresponding to this theorem to be the generation of Hilbert space and quantum mechanics within the singularity as described on pages 9: The active creation of Hilbert space and page 10: The emergence of quantum mechanics.
We describe the role of gravitation in providing energy to kinematic stationary states selected by quantum mechanics on page 17: Gravitation and quantum theory—in the beginning.
Principle 13: The information carried by a physical state is equal to the entropy of the space that it occupies
Shannon's mathematical theory of communication shows that we can make communication error free by coding our messages into packets that are so far apart in message space that the probability of their confusion is negligible. Shannon sought the limits of error free communication over noiseless and noisy channels. The theory he developed is now well known and lies at the heart of communication networks worldwide. Claude E Shannon (1948): A Mathematical Theory of Communication
The validity of these strategies is illustrated by our current ability to send gigabytes of information error free over noisy channels like phone lines. The quantization of communication at the microscopic level supports the hypothesis that our world is a communication network that has evolved to resist error. Wojciech Hubert Zurek: Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical.
Shannon explains that a system that transmits without errors at the limiting rate C predicted by his theorems is an ideal system. Some features of an ideal system are:
1. To avoid error there must be no overlap between signals representing different messages, They must, in other words, be orthogonal, as are the eigenfunctions of a quantum mechanical observable.
2. Such ‘basis signals’ may be chosen at random in the signal space, provided only that they are orthogonal. The same message may be encoded into any satisfactory basis provided that the transformations (codecs) used by used by the transmitter and receiver to encode the message into the signal and decode the signal back to the message are inverses of one another. Codec - Wikipedia
3. The signals transmitted by an ideal system have maximum entropy and are indistinguishable from noise. The fact that a set of physical observations looks like a random sequence is not therefore evidence for meaninglessness. Until the algorithms used to encode and decode such a sequence are known, nothing can be said about its significance.
4. Only in the simplest cases are the mappings used to encode and decode messages linear and topological. For practical purposes, however, they must all be computable with available machines.
5. As a system approaches the ideal, the length of the transmitted packets, the delay at the transmitter while it takes in a chunk of message for encoding, and the corresponding delay for decoding by the receiver, increase indefinitely.
Shannon showed that messages must be encoded in packets, that is quantized, to avoid error. Conversely, an unmodulated continuous signal is incapable of transmitting information.
The key to Shannon's work is the information theoretic definition of information which assigns a measure of information to a symbol proportional to its weight in the entropy of the space of symbols from which the symbol is chosen. Alexandr Yakovlevich Khinchin (1957): Mathematical Foundations of Information Theory, Entropy (information theory) - Wikipedia
Principle 14: Quantization is a certainty principle, not an uncertainty principle
Leopold Kronecker (1823 - 1891) stated in a lecture in 1886 Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers, all else is human work). The quantization of the Universe seems to support this contention. The fundamental interpretation of the quantum on this site is as a logical operator which converts some proposition p into another proposition not-p. In quantum mechanical terms it is the orthogonality operator creating and annihilating or vice-versa. It is the fundamental measure of action in the elementary world.
Max Planck discovered that it is the constant of proportionality between inverse time (frequency) and energy, represented by the equation E = ℏω. Many think it is a source of quantum fluctuations and therefore of energy. Field theoretical calculations based on this interpretation yield values of the cosmological constant that differ from actual measurement by a factor of about 10100. This problem may arise from the confusion of the kinematic role of action in Hilbert space with its dynamical role in interactions between particles in Minkowski space. Steven Weinberg (2000): The Cosmological Constant Problems
Principle 15: An unmodulated continuum carries no information
This principle is a corollary of principle 10: no marks, no information. The nineteenth century continuum, based on infinite concentrations of isolated points, is a mathematical ideal not represented in nature. The notion that elementary particles have zero size leads to the divisions by zero and infinities which have plagued quantum field theories since the late twenties of the last century.
Principle 16: Requisite variety: the measure of information is the measure of control
Turing and Gödel established absolute formal limits to the computability and completeness of mathematics. There are many fine divisions within these limits.
The theory of computability studies what can and cannot be done with a a machine given a certain clock rate, memory size and word length. A major division lies between processes whose execution times vary polynomially and exponentially as functions of the size of the problem. A fundamental question in this areas is known as the P versus NP problem. This is relevant to the question of evolution as explained in Principle 5 above.
Gregory Chaitin has explored the relationship between Gödel's incompleteness theory and the cybernetic principle of requisite variety. Gregory J. Chaitin (1982): Gödel's Theorem and Information
Chaitin writes:
Gödel's theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.
A theorem is an example of formal logical control which demands that the conclusion follows from the axioms or hypotheses. A consequence of requisite variety exploited on this site is that an omnipotent initial singularity with no information can have no control over its actions and so must act randomly. This idea is discussed in more detail on page 5: God's ideas, cybernetics and singularity.
Principle 17: Theology is the comprehensive theory of everything derived from physical data
Theology is discussion about creation and the creator, theos in Greek, god in English, the source of the reality introduced in Principle 1, the initial singularity.
The foundation of theology is something like Descartes' cogito: doubting my existence proves to me that I exist. This leads to an old Scholastic question: quid est hoc quod est esse? what is it to be? A simple, satisfactory and common answer is that I am a child of my ancestors. My ancestors are my source and they define me.
Traditional theology often assumes that the creator existed before the world began and knew exactly what they were going to do when they decided to make it. The creation of Western Christian theology begins with Genesis. They authors of Genesis replaced the Sumerian Gods with their own single God and told their story twice using different names for God, Elohim (generic) and Yahweh (personal). Genesis creation narrative - Wikipedia, Sumerian religion - Wikipedia
On this site my primordial ancestor is the simplest structure imaginable, modelled on the absolute simplicity of the Christian God. I begin with the assumption that the quantum of action is a thing, identical to the Christian God, which we first meet in the initial singularity implied by general relativity. Both the initial singularity and the Christian God share three properties, they exist, they are absolutely simple, and they are the source of the Universe. Hawking & Ellis (1975): The Large Scale Structure of Space-Time
Because the universe started so simple, the explanation of the current structure must be simple. This idea is expressed in three axioms:
1. Pure action, by self reference, can try anything. An ancient Christian expression of this idea is that God's image of themself is identically god, the second person of the Trinity.
2. The cybernetic principle of requisite variety says such a simple beginning cannot control its future, so primordial events are random sequences.
3. Only those sets of events that form a group can maintain themselves since the interaction of any two members of a group produces another member of the group.
Given that the Universe is divine and visible to everybody, theology can then become a real science
Copyright:
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Notes and references
Further reading
Books
Ashby (1964), W Ross, An Introduction to Cybernetics, Methuen 1956, 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics.'
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Cantor (1897, 1955), Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1895, 1897, 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.'
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Chaisson (2002), Eric J., Cosmic Evolution: The Rise of Complexity in Nature, Harvard University Press 2002 ' In Cosmic Evolution Chaisson addresses some of the most basic issues we can contemplate: the origin of matter and the origin of life, and the ways matter, life, and radiation interact and change with time. Guided by notions of beauty and symmetry, by the search for simplicity and elegance, by the ambition to explain the widest range of phenomena with the fewest possible principles, Chaisson designs for us an expansive yet intricate model depicting the origin and evolution of all material structures. He shows us that neither new science nor appeals to nonscience are needed to understand the impressive hierarchy of the cosmic evolutionary story, from quark to quasar, from microbe to mind.'
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Darwin (1875, 1998), Charles, and Harriet Ritvo (Introduction), The Variation of Animals and Plants Under Domestication (Foundations of Natural History), Johns Hopkins University Press 1875, 1998 ' "The Variation, with its thousands of hard-won observations of the facts of variation in domesticated species, is a frustrating, but worthwhile read, for it reveals the Darwin we rarely see -- the embattled Darwin, struggling to keep his project on the road. Sometimes he seems on the verge of being overwhelmed by the problems he is dealing with, but then a curious fact of natural history will engage him (the webbing between water gun-dogs' toes, the absurdly short beak of the pouter pigeon) and his determination to make sense of it rekindles. As he disarmingly declares, 'the whole subject of inheritance is wonderful.'.
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Feynman (2002), Richard, Feynman Lectures on Gravitation, Westview Press 2002 ' 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. '
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Fortun (1998), Mike, and Herbert J Bernstein, Muddling Through: Pursuing Science and Truths in the Twenty-First Century, Counterpoint 1998 Jacket: ' Messy. Clumsy. Volatile. Exciting. These words are not often associated with the science, which for most people still connote exactitude, elegance, reliability and a rather plodding certainty. But the real story is something quite different. The sciences are less about the ability to know and to control than they are about the unleashing of new forces, new capacities for changing the world. The sciences as practised exist not in some pristine world of "objectivity," but in what Mike Fortnum and Herbert Bernstein call "the Muddled Middle".
This book explores the way science makes sense of the world and how the world makes sense of science. It is also about politics and culture—how these forces shape the sciences and are shaped by them in return.'
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Galilei (1610, 1957), Galileo, and Stillman Drake (translator), Discoveries and Opinions of Galileo: Including the Starry Messenger (1610 Letter to the Grand Duchess Christina), Doubleday Anchor 1957 Amazon: 'Although the introductory sections are a bit dated, this book contains some of the best translations available of Galileo's works in English. It includes a broad range of his theories (both those we recognize as "correct" and those in which he was "in error"). Both types indicate his creativity. The reproductions of his sketches of the moons of Jupiter (in "The Starry Messenger") are accurate enough to match to modern computer programs which show the positions of the moons for any date in history. The appendix with a chronological summary of Galileo's life is very useful in placing the readings in context.' A Reader.
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Hawking (1975), Steven W, and G F R Ellis, The Large Scale Structure of Space-Time, Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.'
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Khinchin (1957), Aleksandr Yakovlevich, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.'
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Lonergan (1992), 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'
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Links
Action at a distance - Wikipedia, Action at a distance - Wikipedia, the free encyclopedia, ' In physics, action at a distance is the concept that an object can be moved, changed, or otherwise affected without being physically touched (as in mechanical contact) by another object. That is, it is the nonlocal interaction of objects that are separated in space. This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant objects. For example, Coulomb's law and Newton's law of universal gravitation are such early theories. More generally "action at a distance" describes the failure of early atomistic and mechanistic theories which sought to reduce all physical interaction to collision. The exploration and resolution of this problematic phenomenon led to significant developments in physics, from the concept of a field, to descriptions of quantum entanglement and the mediator particles of the Standard Model.' back |
Alan Turing (1936), On Computable Numbers, with an application to the Entscheidungsproblem, 'The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by some finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable of a real or computable variable, computable predicates and so forth. . . . ' back |
Albert Einstein (1915), The Field Equations of Gravitation, ' In two recently published papers I have shown how to obtain field equations of gravitation that comply with the postulate of general relativity, i.e., which in their general formulation are covariant under arbitrary substitutions of space-time variables. . . . With this, we have finally completed the general theory of relativity as a logical structure. The postulate of relativity in its most general formulation (which makes space-time coordinates into physically meaningless parameters) leads with compelling necessity to a very specific theory of gravitation that also explains the movement of the perihelion of Mercury. However, the postulate of general relativity cannot reveal to us anything new and different about the essence of the various processes in nature than what the special theory of relativity taught us already. The opinions I recently voiced here in this regard have been in error. Every physical theory that complies with the special theory of relativity can, by means of the absolute differential calculus, be integrated into the system of general relativity theory-without the latter providing any criteria about the admissibility of such physical theory.' back |
Apophatic theology - Wikipedia, Apophatic theology - Wikipedia, the free encyclopedia, 'Apophatic theology (from Greek ἀπόφασις from ἀπόφημι - apophēmi, "to deny")—also known as negative theology or via negativa (Latin for "negative way")—is a theology that attempts to describe God, the Divine Good, by negation, to speak only in terms of what may not be said about the perfect goodness that is God. It stands in contrast with cataphatic theology.' back |
Aquinas, Summa I, 25, 3, Is God omnipotent?, '. . . God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. 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.' back |
Aquinas, Summa, I, 3, 7, Is God 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, Summa, I, 44, 1, Is God the efficient cause of all things?, ' I answer that, It must be said that every being in any way existing is from God. For whatever is found in anything by participation, must be caused in it by that to which it belongs essentially, . . . Therefore all beings apart from God are not their own being, but are beings by participation. Therefore it must be that all things which are diversified by the diverse participation of being, so as to be more or less perfect, are caused by one First Being, Who possesses being most perfectly.' back |
Axiom - Wikipedia, Axiom - Wikipedia, the free encyclopedia, 'An axiom, or postulate, is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.The word comes from the Greek ἀξίωμα (āxīoma) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning. Axioms define and delimit the realm of analysis; the relative truth of an axiom is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other relative truths. No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be an irrelevant and impossible contradiction in terms.' back |
Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, ' The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then the measured result will be one of the eigenvalues λ of A, and the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'.' back |
Boson - Wikipedia, Boson - Wikipedia, the free encyclopedia, 'In particle physics, bosons are particles with an integer spin, as opposed to fermions which have half-integer spin. From a behaviour point of view, fermions are particles that obey the Fermi-Dirac statistics while bosons are particles that obey the Bose-Einstein statistics. They may be either elementary, like the photon, or composite, as mesons. All force carrier particles are bosons. They are named after Satyendra Nath Bose. In contrast to fermions, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the same place in space.' back |
Claude E Shannon (1948), A Mathematical Theory of Communication, ' The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back |
Claude Shannon (1949), Communication in the Presence of Noise, 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two “function spaces,” and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of “ideal” systems which transmit at this maximum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.' [C. E. Shannon , “Communication in the presence of noise,” Proc. IRE, vol. 37, pp. 10–21, Jan. 1949.] back |
Codec - Wikipedia, Codec - Wikipedia, the free encyclopedia, 'A codec is a device or computer program that encodes or decodes a data stream or signal. Codec is a portmanteau of coder/decoder. . . . IA coder or encoder encodes a data stream or a signal for transmission or storage, possibly in encrypted form, and the decoder function reverses the encoding for playback or editing. Codecs are used in videoconferencing, streaming media, and video editing applications. In the mid-20th century, a codec was a device that coded analog signals into digital form using pulse-code modulation (PCM). Later, the name was also applied to software for converting between digital signal formats, including companding functions. ' back |
Collectively exhaustive events - Wikipedia, Collectively exhaustive events - Wikipedia, the free encyclopedia, ' In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single outcome are collectively exhaustive, because they encompass the entire range of possible outcomes.' back |
Dynamics - Wikipedia, Dynamics - Wikipedia, the free encyclopedia, ' Dynamics is the branch of classical mechanics that is concerned with the study of forces and their effects on motion. Isaac Newton was the first to formulate the fundamental physical laws that govern dynamics in classical non-relativistic physics, especially his second law of motion. ' back |
Entropy (information theory) - Wikipedia, Entropy (information theory) - Wikipedia, the free encyclopedia, 'In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. In this context, a 'message' means a specific realization of the random variable. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication".' back |
Fermion - Wikipedia, Fermion - Wikipedia, the free encyclopedia, 'In particle physics, fermions are particles with a half-integer spin, such as protons and electrons. They obey the Fermi-Dirac statistics and are named after Enrico Fermi. In the Standard Model there are two types of elementary fermions: quarks and leptons. . . . In contrast to bosons, only one fermion can occupy a quantum state at a given time (they obey the Pauli Exclusion Principle). Thus, if more than one fermion occupies the same place in space, the properties of each fermion (e.g. its spin) must be different from the rest. Therefore fermions are usually related with matter while bosons are related with radiation, though the separation between the two is not clear in quantum physics. back |
Feynman, Leighton & Sands FLP III_17, Chapter 17: Symmetry and Conservation Laws, ' In classical physics there are a number of quantities which are conserved—such as momentum, energy, and angular momentum. Conservation theorems about corresponding quantities also exist in quantum mechanics. The most beautiful thing of quantum mechanics is that the conservation theorems can, in a sense, be derived from something else, whereas in classical mechanics they are practically the starting points of the laws. . . . In quantum mechanics, however, the conservation laws are very deeply related to the principle of superposition of amplitudes, and to the symmetry of physical systems under various changes.' back |
Feynman, Leighton & Sands FLP III:09, Chapter 9: The Ammonia Maser, 'In this chapter we are going to discuss the application of quantum mechanics to a practical device, the ammonia maser. You may wonder why we stop our formal development of quantum mechanics to do a special problem, but you will find that many of the features of this special problem are quite common in the general theory of quantum mechanics, and you will learn a great deal by considering this one problem in detail. The ammonia maser is a device for generating electromagnetic waves, whose operation is based on the properties of the ammonia molecule which we discussed briefly in the last chapter.' back |
Feynman, Leighton and Sands FLP II_02, Chapter 2: Differential Calculus of Vector Fields, ' What it means really to understand an equation—that is, in more than a strictly mathematical sense—was described by Dirac. He said: “I understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it.” So if we have a way of knowing what should happen in given circumstances without actually solving the equations, then we “understand” the equations, as applied to these circumstances. A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist. ' back |
Genesis creation narrative - Wikipedia, Genesis creation narrative - Wikipedia, the free encyclopedia, ' The authors of the Hebrew creation narrative borrowed themes from Mesopotamian mythology, but adapted them to their unique belief in one God. The first major comprehensive draft of the Pentateuch (the series of five books which begins with Genesis and ends with Deuteronomy) is thought to have been composed in the late 7th or the 6th century BCE (the Jahwist source) and was later expanded by other authors (the Priestly source) into a work very like Genesis as known today. The two sources can be identified in the creation narrative: Priestly and Jahwistic. The combined narrative is a critique of the Mesopotamian theology of creation: Genesis affirms monotheism and denies polytheism.' back |
Gregory J. Chaitin (1982), Gödel's Theorem and Information, 'Abstract: Gödel's theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.' |
Hebrew Bible - Wikipedia, Hebrew Bible - Wikipedia, the free encyclopedia, ' The Hebrew Bible . . . is a term referring to the books of the Jewish Bible as originally written mostly in Biblical Hebrew with some Biblical Aramaic. The term closely corresponds to contents of the Jewish Tanakh and the Protestant Old Testament (see also Judeo-Christian) but does not include the deuterocanonical portions of the Roman Catholic or the Anagignoskomena portions of the Eastern Orthodox Old Testaments. The term does not imply naming, numbering or ordering of books, which varies (see also Biblical canon).' back |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Kinematics - Wikipedia, Kinematics - Wikipedia, the free encyclopedia, 'Kinematics (from Greek . . . kinein, to move) is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.' back |
P versus NP problem - Wikipedia, P versus NP problem - Wikipedia, the free encyclopedia, ' The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified (technically, verified in polynomial time) can also be solved quickly (again, in polynomial time). The underlying issues were first discussed in the 1950s, in letters from John Forbes Nash Jr. to the National Security Agency, and from Kurt Gödel to John von Neumann. The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper " The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field.' back |
Peter Osper (1957), Review: Albert Szent-Györgyi (1957): Bioenergetics, ' Everyone who is interested in biological chemistry will want to read and reread this book, and then design some experiments to prove Szent-Györgyi: right or wrong. One gets the impression that Szent-Györgyi will not be too unhappy to be proved wrong. . . .' In 1957 the scientist Albert Szent-Györgyi released this book which contained a part titled “Biological Structures and Functions”. The following statement without attribution was employed as an epigraph for this part (page 56): https://archive.org/details/bioenergetics00szen/page/57/mode/1up “Research is to see what everybody has seen and think what nobody has thought.” back |
Richard Feynman, Lectures on Physics III:17 Symmetry and Conservation Laws, 'The most beautiful thing of quantum mechanics is that the conservation theorems can, in a sense, be derived from something else, whereas in classical mechanics they are practically the starting points of the laws. . . . In quantum mechanics, however, the conservation laws are very deeply related to the principle of superposition of amplitudes, and to the symmetry of physical systems under various changes. This is the subject of the present chapter. Although we will apply these ideas mostly to the conservation of angular momentum, the essential point is that the theorems about the conservation of all kinds of quantities are—in the quantum mechanics—related to the symmetries of the system.' back |
Steven Weinberg (2000), The Cosmological Constant Problems, 'Abstract. The old cosmological constant problem is to understand why the vacuum energy is so small; the new problem is to understand why it is comparable to the present mass density. Several approaches to these problems are reviewed. Quintessence does not help with either; anthropic considerations offer a possibility of solving both. In theories with a scalar field that takes random initial values, the anthropic principle may apply to the cosmological constant, but probably to nothing else.' back |
Sumerian religion - Wikipedia, Sumerian religion - Wikipedia, the free encyclopedia, ' Sumerian religion was the religion practiced by the people of Sumer, the first literate civilization of ancient Mesopotamia. The Sumerians regarded their divinities as responsible for all matters pertaining to the natural and social orders. . . . The Sumerians believed that the universe had come into being through a series of cosmic births such as gods. First, Nammu, the primeval waters, gave birth to Ki (the earth) and An (the sky), who mated together and produced a son named Enlil. Enlil separated heaven from earth and claimed the earth as his domain. Humans were believed to have been created by AnKi or Enki, the son of the An and Ki.' back |
Symmetry (physics) - Wikipedia, Symmetry (physics) - Wikipedia, the free encyclopedia, ' n physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group).' back |
Unitarity (physics) - Wikipedia, Unitarity (physics) - Wikipedia, the free encyclopedia, ' In quantum physics, unitarity means that the sum of probabilities of all possible outcome of any event is always 1. This is necessary for the theory to be consistent. This implies that the operator which describes the progress of a physical system in time must be a unitary operator. This operator is e iHt where H is the Hamiltonian of the system and t is [an increasing number, not necessarily time since we are in Hilbert space where there is no space-time].' back |
Wojciech Hubert Zurek (2008), Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical, 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3)) Measurements transfer information about a system to the apparatus, and then further on – to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide framework for the “wavepacket collapse”, designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates.' back |
Zero-energy universe - Wikipedia, Zero-energy universe - Wikipedia, the free encyclopedia, 'The zero-energy universe hypothesis proposes that the total amount of energy in the universe is exactly zero: its amount of positive energy in the form of matter is exactly cancelled out by its negative energy in the form of gravity. . . . The zero-energy universe theory originated in 1973, when Edward Tryon proposed in the journal Nature that the universe emerged from a large-scale quantum fluctuation of vacuum energy, resulting in its positive mass-energy being exactly balanced by its negative gravitational potential energy.' back |