natural theology

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Notes

Sunday 2 May 2021 - Saturday 8 May 2021

[Notebook: DB 86: Hilbert / Minkowski]

[page 210]

Sunday 2 May 2021

Politics vs science → personal desires vs transpersonal realities. Wang Xiangwei: China’s treatment of Oscar winner Chloé Zhao shows how to lose friends and not influence people

Streater and Wightman work to sanitize continuous quantum theory and continuous field theory by introducing the mathematical theory of distributions to bring Dirac's delta, essential to the traditional mapping of quantum theory to continuous space-time, out of the ad hocery of physics and into the fold of respectable mathematics. To me, however, their good work seems to emphasize even further the inappropriateness of continuous mathematics for the description of the quantum world. Distribution (mathematics) - Wikpedia, Streater & Wightman: PCT, Spin, Statistics and All That

[page 211]

Thoroughgoing quantization seems to be the way but how do we manage it to get numerical results that match observation? Here we come across many difficulties and it is tempting to endorse the practical experiments and mathematical methods of physics while changing our interpretation of what is going on. All quantum measurements are example of binning and counting discrete events and at this level it is the application of the law of large numbers to sets of events that justifies the use of continuous real mathematics. The application of complex amplitudes, however, is a different question, and the fact that complex numbers are idea for discussing periodic functions and their superposition is a strong case for continuous mathematics. Nevertheless the fact that computation, mechanical computation, is a periodic process controlled by clocks and synchronization is a suitable explanation for the wave function approach to the theory. The complex exponential miraculously simplifies the modelling of all periodic phenomena, so it applies to mathematical and physical computation equally. We use memory and buffering to synchronize the interaction of individual machines in a computer network. Nature, at its most fundamental level may not have this technology so it must rely for much of its [real time] communication by only favouring connections whose frequencies of processing are closely matched, as in the interactions between photons and orbiting electrons and, I suspect, in all other interactions in which gauge particles, phase and frequency [energy] are the determinants of the rate of interaction.

Monday 3 May 2021

Von Neumann took the first steps toward developing an axiomatic foundation for quantum theory by defining abstract Hilbert space and developing some of its properties. Streater and Wightman propose 5 axioms for the notions of field and field theory. John von Neumann: Mathematical Foundations of Quantum Mechanics

[page 212]

Streater page 97: '0. The states of the system are described by [correspond to] unit rays in a separable Hilbert space. Relativistic transformation of the states is given by continuous unitary representations of the inhomogeneous [group] SL(2, C).' Do not like this as it assumes that the states are in Minkowski space which seems to place unnecessary constraints on them and makes it difficult to understand entanglement and spooky action at a distance, which is real even if it cannot be used for the deterministic transmission of information. Special linear group - Wikipedia

' There is an invariant state Ψ0, the vacuum, such that U(a, A)Ψ0 = Ψ0.' Where did that come from? As in Zee, the vacuum seems to be the deus in machina that makes field theory work [the matter on which the formal fields are built]. We would like to think that it is the random activity of the initial singularity, the quantum of action whose nature is to act. The first step in coding up a logical theory is to produce a model of this initial vacuum. Anthony Zee (2010): Quantum Field Theory in a Nutshell

page 98: 'I Assumptions about the Domain and Continuity of the Field

'For each test function defined on f ∈ S defined in space-time there exists a set ψi(f) . . .ψn(f) of operators. These operators together with their adjoints ψi(f)* . . .ψn(f)* are defined on a domain D of vectors dense in H. Furthermore D is a linear set containing Ψ0 . . . '

'II Transformation Law of the Field

page 100: 'III Local Commutativity, Sometimes called Microscopic Causality'

These assumptions define a field but not a field theory. ' To be a field theory a relativistic quantum theory must have enough fields so its states can be uniquely characterized using fields and function of fields.'

[page 213]

Streater page 102: ' 'IV Asymptotic Completeness H = H in = H out

The theory I dream of has its foundations in quantum computation and quantum communication whose roots lie beneath Minkowski space. Part of the dream is that by the end of the year we will have brought this idea out into the open. From an entropic point of view, we imagine that a separable Hilbert space can represent 0 states, which, with the help of Cantor, can create a transfinite logical network that provides an address for every quantum of action in the universe. See Prolegomena for a prospectus. Jeffrey Nicholls (2019): Prolegomenon to Scientific Theology

Rovelli: Relational Quantum Mechanics. Carlo Rovelli

We assume that relations are established by communication which is underpinned by computation, so Rovelli's interpretation is closely related to the ideas of the transfinite quantum network, aka field theory.

Rovelli page 1: 'This paper is based on a critique of a notion generally assumed uncritically. As such it bears a vague resemblance to Einstein's discussion of special relativity which is based on a critique of the notion of absolute simultaneity. The notion rejected here is the notion [much beloved by Einstein] of absolute or observer-independent values of physical quantities.' Einstein, Podolsky and Rosen: Can the Quantum Mechanical Description of Physical Reality be Considered Complete?

Because

'the empirical evidence at the basis of quantum mechanics forces us to accept that distinct observers give different descriptions of the same events.' This is a universal observation at all scales and makes it very difficult to describe what we mean by 'the same events' when every event is a unique concrete example of two systems meeting and observing one another.

[page 214]

' The issue is not to replace or fix [quantum mechanics] but rather to understand what precisely it says about the world; or equivalently: what precisely have we learnt from experimental micro-physics' [for my money that it is pretty much exactly the same as human relationships and other relationships at all scales from nuclear micro to galactic macro, what I call symmetry with respect to complexity, founded on Cantor's theorem.]. Cantor's theorem - Wikipedia

' . . . the fact that no interpretation has so far succeeded in convincing the majority if physicists indicates, I believe, that the problem of the interpretation of quantum mechanics has not been fully disentangled yet.

Rovelli page 2: '. . . quantum mechanics will cease to look puzzling only when we will be able to derive the formalism of the theory from a set of simple physical assertions ("postulates", "principles") about the world. Therefore we should not try to append a reasonable interpretation to the quantum mechanics formalism, but rather derive the formalism from a set of experimentally motivated postulates.

For example:

' The formal content of special relativity is coded into the Lorentz transformations written by Lorentz, not by Einstein, and before 1905. So what was Einstein's contribution? It was to understand the physical meaning of the Lorentz transformations. . . . Einstein's contribution to special relativity has been the interpretation of the theory, not its formalism.

' The physical interpretation proposed by Lorentz himself was a physical contraction of moving bodies caused by a complex and unknown electromagnetic interaction between the moving bodies and the ether. It was quite an unattractive explanation, remarkably similar to certain

[page 215]

wave function collapse presently investigated!

' The aim of this paper is to hunt for [the incorrect notion behind our unease wth quantum mechanics] with the hope that by exposing it clearly to public contempt we could . . . fully understand what does the theory assent about the world.

' Einstein was so persuasive with his interpretation of the Lorentz equations because he did not append an interpretation to them: rather he rederived them, starting from two postulates with terse physical content — equivalence of inertial observers, and universality of the speed of light.'

' I have a methodological suggestion for the problem of interpretation of quantum mechanics. Finding the set of physical facts from which the quantum mechanics formalism can be derived.' ie communication and computation.

Rovelli page 3: ' II Quantum Mechanics is a theory about Information' or it might be better to say communication, information on the move.

page 4: 'Main observation: In quantum mechanics different observers may give different accounts of the same sequence of events.'

Hyopthesis 1: 'All systems are equivalent, macroscopic ≡ microscopic

page 5: Objection 2: '. . . quantum state is a fictitious non-physical mental construction; the physical content of the theory is given in the outcomes of measurements.'

[page 216]

Rovelli page 6: after some objections (which do not defeat the main observation) we go to 'C. Main discussion: - observer independence is dead, "state", "value of variable" or "outcome of measurement" are relational notions in the same sense in which velocity is relational in classical mechanics.

page 7: 'Hyothesis 2: (Completeness) so

'Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world.'

so the sought for incorrect notion is that '. . . the notion of true, universal, observer independent description of the state of the world. If the notion of observer-independent description of the world is unphysical, a complete description of the world is exhausted by the relevant information that systems have about each other. Namely that there is nether an absolute state of the system, nor absolute properties that the system has at a certain time.' ie every interaction is a communication that affects both participants ontologically. As Thomas notes relations, communications are real in god. Aquinas, Summa I, 28, 1: Are there real relations in God?

D. Relation between descriptions

Tuesday 4 May 2021

Quantum mechanics has time / serial output (?)

[page 217]

Rovelli page 8: 'The absolute state of affairs in the world is a meaningless notion' Silly, it is the concrete content of a particular communication event, that is any event. "Do you take me to be your partner?" Yes/No. "Do you take A?" Yes. "Do you take B?" No, etc.

Once uttered (observed) these answers become part of the classical world and can be recorded and stored in a register for all classical people to see regardless of their mental / quantum mechanical states.

page 10. 'On the reconstruction of quantum mechanics.'

'Since information is discrete, any process of acquisition of information can be decomposed into the [serial ?] acquisition of elementary bits of information.'

'A yes/no measurement is represented by a projection operator onto a subset of Hilbert space . . . ' Projection (linear algebra) - Wikipedia

'In place of the notion of state, which refers solely to the system, the notion that the information that a system has about another system has been introduced.' Which is a measure of the state of the system. When I ask if you will marry me, your answer reveals your state.

page 11: ' The two main postulates'

Postulate 1: (Limited information) There is a maximum amount of relevant information that can be extracted from a system.

page 12: ' Postulate 2: (Unlimited information) It is always possible to acquire new information about a system.'

'C. Reconstruction of the formalism, and the third postulate.'

page 14: Postulate 3: (Superposition principle)

[page 217a]

Rovelli page 14: ' D. The Observer observed '

'. . . let me stress again that I believe that the common mistake in analyzing measurement issues in quantum mechanics is to forget that two can compare their information (their measurement outcomes) only by physically interacting with each other. This means that there is no way to compare the "information possessed by O" with the "information possessed by P" without considering a quantum physical interaction, or a quantum measurement between the two.'

Maybe this explains why specific information cannot be transmitted by entanglement. When I observe my element of an entangled pair and get the result 0, I know that the observation of the other element of the pair will be 1, and vice versa, but I cannot be certain [in advance] what I will observe, and so cannot be certain what the holder of the other half of the entanglement will receive, so at best I am sending random signals, nevertheless very efficiently!

page 15: 'IV Critique of the concept of state'

'A. Any observation requires an observer', the fundamental concrete tautology.

' If different observers give different descriptions of the state of the same system, this means that the notion of state is observer dependent'. Since the act of observation changes the state of both observer and observed [eg player and die] and two observers cannot observe a system simultaneously, this is a tautological conclusion.

[page 217b]

Rovelli page 17: 'According to the point of view described here the Schrödinger unitary evolution of the system S breaks down simply because the system interacts with something that is not taken into account in the evolution equations. Unitary evolution requires the system to be isolated which is exactly what ceases to be true during the measurement due to the interaction with the observer. If we include the observer into the system, then the evolution is still unitary, but we are now dealing with the description of a different observer.' The unobserved universe as a whole must therefore be unitary, rotating like a cosmic planet.

Breuer (1994) ' No system (quantum or classical) can perform complete self-measurement.'

Bohr distinguished to worlds, the quantum and the classical, and understood the classical world to be observing the quantum world. From a practical point of view, this is how we still work. Quantum mechanics explains transistors and laser and then engineers manipulate the composition and configuration of the materials necessary to make the quantum systems work, guided by theory and endless trial and error. As Rovelli makes clear, however, the theoretical interface between the two worlds is far from settled. [continued tomorrow at **]

Quantum mechanics is very similar to an orchestra, a linear superposition of instruments, each of which has an eigenfunction which can be resolved by acute hearing [and which taken together] create an effect ensemble which is, we might say, classical.

Wednesday 5 May 2021

[page 218]

Rovelli page 14: ** Rovelli states in D. The observer observed ' We now have the full formal machinery of quantum mechanics with an interpretative novelty: the absence of the notion of [the (?)] state [of (?)] the system.' He begins by attempting to derive the quantum formalism from the postulates 1 (limited) and 2 (unlimited) information. ' The motivation of the second postulate is fully experimental. We know that all quantum systems (and all systems are quantum systems) have the property that if we know their quantum state |ψ> exactly we can still "learn" something about them by performing a measurement of a quantity O such that |ψ> is not an eigenstate of O. Since the amount of information that O can have about S is limited by postulate 1, some of the old relevant information becomes irrelevant. In particular, if a new question Q (not determined by the previous information gathered) is asked, then O loses (at least) one bit of previous information. So that after asking the question Q, new information is available, but the total amount of relevant information that the system has does not exceed N bits.'

'Rather surprisingly, these two postulates are (almost) sufficient to construct the full formalism of quantum mechanics. Namely, one may assert that the physics content of the general formalism of quantum

[page 219]

mechanics is (almost) nothing but a series of consequences of the two physical facts expressed in postulates 1 and 2.'

so we go to 'C. Reconstruction of the formalism, and the third postulate

Rovelli page 12: First postulate set of question is an "orthomodular lattice". Complemented lattice - Wikipedia

We estimate the maximum information in W(S) [the set of questions that can be asked to a system S] by finding an ensemble of N independent questions whose answers for a string of N bits which encode 2N possible outcomes as binary numbers.

page 13: This ensemble of "complete questions" may not be unique but nevertheless it seems to be assumed that all ensembles of complete questions yields the same number N of bits of information.

Now postulate 2, unlimited information. To what extent does a complete set of questions determine the outcome of an additional question (asked with a new measurement operator with different eigenfunctions)?

The result: ' Postulate 3 (superposition principle). If c and b define two complete families of questions' then a unitary matrix exists which which relates ic and b by a unitary change of basis.

So we do not seem to have got anywhere interesting and fall back to von Neuman's proof that observation increases entropy, somehow breaking Rovelli's argument which says that different sets of questions (measurement operators) preserve unitarity.

[page 220]

Dear Prof Rovelli (again). Your book Helgoland inspired me to read your article Relational quantum mechanics, which led me to the following thoughts:

1. Penrose, Hawking and Ellis see in Einstein's general theory a suggestion that the universe began from a structureless point comprising all the energy of the universe that became a 'big bang' that created the universe within itself. Hawking & Ellis: The Large Scale Structure of Space-Time

2. Aquinas, following Aristotle, found that god is also a structureless necessary being whose essence is to be. We might identify god and the initial singularity on the grounds that both exist, are absolutely simple and are sources of the universe. Aquinas, Summa, I, 3, 7: Is God altogether simple?

3. Since it has no structure the entropy of the initial singularity must be zero. The cybernetic principle of requisite variety requires that a controller must have entropy equal or greater than the system controlled. Since the entropy of the initial singularity is zero, it has no power over the structure of the universe which nevertheless, currently, has very high entropy. Ashby (1964)

4. Determinism cannot increase entropy. An isolated quantum system evolving unitarily evolves at constant entropy, like a reversible entropy preserving codec. Codec - Wikipedia

5. Von Neumann shows that [quantum] observation as he understands it increases entropy and so we are led to see it as the source of the creation. John von Neumann: Mathematical Foundations of Quantum Mechanics chapter 5.

[page 221]

of the universe in quantum interactions.

6. As in the theory of evolution the creativity is to be found in the randomness inherent in reproduction. We may imagine that the initial singularity only reproduces itself by random quantum actions and we therefore conclude that the initial singularity, identical to the classical god, acts according to quantum mechanics to create the world.

7. Your approach to quantum mechanics in Relational Quantum Mechanics seems to reduce it to unitary determinism, since the recursive process of observers observing observers appear to be unitary, so that we are led to the [the conclusion that] the unobserved universe must be unitary, deterministic and unable to proceed beyond its initial state.

Friday 7 May 2021

8. The most important thing, from a theological point of view is the ability of quantum theory to explain creation, that is increase in entropy. This cannot be done with either classical mechanics or the 'classic' unitary approach to quantum mechanics. The answer must lie in the principle of requisite variety and the inability of the low entropy past to control the future. Since "all systems are quantum systems" any quantum theory must embrace the fact that entropy increases. We may look for ideas here in classical work based on Boltzmann's H-theorem.

Where to now? The critical feature of quantum mechanics for this

[page 222]

project is its ability to create entropy, that is to create new states. From this point of view Rovelli's derivation appears to fail and the need for the operators used in quantum computation and quantum communication to respect unitarity limits the usefulness of this theory. The approach that seems forced on me has been simmering in the back of my mind for a long time. [It] is to develop quantum mechanics in a concrete way from what I have called the reification of the quantum of action and the notion that the essence of the quantum of action is to act, but requisite variety says that it cannot control the consequences of its action any more than parents can control the lives of their children. The next step is to write this into the introduction to quantum-theology and begin developing the website in detail to see where it leads, building on the idea that the principal role attributed to god is creator, and so we wish to see how a divine universe can create itself from pure action.

God is love - god is action

Saturday 8 May 2020

For a long time I have been writing that theology and religion go together like science and technology, like biology and medicine. The acid test for technology is that it works, a roof that keeps out the rain, a phone at the end of a long string of complex systems that displays a radar image of the state of the atmosphere any any point on Earth. I wake up in the morning because all the atoms and molecules (80 kg of them) that contribute to my life are synchronized for life. One day this synchronicity will cease and I

[page223]

will be dead. Theology, as a theory of everything, attempt to explain how everything works For a long time we have lived with very little knowledge of this. In the old days we attributed all this wonderful functioning to a god, omniscient and omnipotent, which kept everything ticking along and was also responsible for the occasional failures. Our experience of human social relationships often led us to attribute such failures to bad actors [sinners] and divine punishment for individual or collective bad actions. The whole Christian story, known as the History of Salvation fits into this picture, which is largely false. The first step out of this view was the invention of evidence based justice which was founded on the notion of innocence until guilt is proven by some sort of evidence. Science extends the paradigm to god. We do not accept that god is vindictive when we understand the mechanisms of disasters such as cyclones and tidal waves and see that they are just extreme examples of the normal functioning of the divine system. So now that we have a pretty comprehensive picture of how the world works we can built a new science and technology of the inner workings of the divinity and use this knowledge to guide our behaviour. No longer is it effective to pray to a mysterious god to make things peaceful, we know how things work and we can act accordingly. There is no point in praying for peace, we can make it by designing vaccines, by adjusting our impacts on the planet to enable it to function optimally and so on. All this is a matter of collective knowledge and action controlled by politics. We know how to make politics go well, and we know what makes it go bad, and the aim of natural theology is to propagate this knowledge and make everyone aware of its benefits. Anshel Pfeffer: Opinion | Does Anyone Really Care About the Haredim?, Acemoglu & Robertson (2012): Why Nations Fail: The Origins of Power, Prosperity and Poverty

Fisher. The root of evil is the fact of evolution constrained by zero

[page 224]

processes that mean if I am to have you must go without. The root of good is the creation that expands the pool of resources so that everyone can have. The potential evil in this approach is that we will exploit the Earth beyond its capacity. The root of this problem is that it is not our problem, god will provide. The political root of this evil is that the rich and powerful are happy with their lot and see no reason to stop exploiting the poor and the planet in order to maintain their status. Max Fisher: ‘Belonging Is Stronger Than Facts’: The Age of Misinformation

Hilbert space served as a space of quantum mechanical possibility like Minkowski space-time. We have to fill in the details using empirical observations and we require that the space rule none of these out.

We begin with a divine initial singularity and develop the trinity using a toy quantum field theory going through energy time and bosons to potential, space, fermions, momentum, and explain periodicity by computation.

De Gaulle De Gaulle (film) - Wikipedia

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Further reading

Books

Acemoglu (2012), Daron, and James Robinson, Why Nations Fail: The Origins of Power, Prosperity and Poverty, Crown Business 2012 "Some time ago a little-known Scottish philosopher wrote a book on what makes nations succeed and what makes them fail. The Wealth of Nations is still being read today. With the same perspicacity and with the same broad historical perspective, Daron Acemoglu and James Robinson have retackled this same question for our own times. Two centuries from now our great-great- . . . -great grandchildren will be, similarly, reading Why Nations Fail." —George Akerlof, Nobel laureate in economics, 2001  
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Aquinas, Thomas, Summa Theologica (translated by Fathers of the English Dominican Province), Tabor Publishing 1981 'Brother Thomas raised new problems in his teaching, invented a new method, used new systems of proof. To hear him teach a new doctrine, with new arguments, one could not doubt that God, by the irradiation of this new light and by the novelty of this inspiration, gave him the power to teach, by the spoken and written word, new opinions and new knowledge.' (William of Tocco, T's first biographer) 
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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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Augustine (419, 1991), and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 399-419, 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augustine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.  
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Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory). 
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Chaitin, Gregory J, Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, World Scientific 1987 Jacket: 'Algorithmic information theory is a branch of computational complexity theory concerned with the size of computer programs rather than with their running time. . . . The theory combines features of probability theory, information theory, statistical mechanics and thermodynamics, and recursive function or computability theory. ... [A] major application of algorithmic information theory has been the dramatic new light it throws on Goedel's famous incompleteness theorem and on the limitations of the axiomatic method. . . .' 
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Chaitin, Gregory J, Algorithmic Information Theory, Cambridge UP 1987 Foreword: 'The crucial fact here is that there exist symbolic objects (i.e., texts) which are "algorithmically inexplicable", i.e., cannot be specified by any text shorter than themselves. Since texts of this sort have the properties associated with random sequences of classical probability theory, the theory of describability developed . . . in the present work yields a very interesting new view of the notion of randomness.' J T Schwartz 
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Chaitin, Gregory J, "Goedel's Theorem and Information" in Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, Reprinted from the International Journal of Theoretical Physics (1982) 22, 941-954., World Scientific 1987 Abstract: 'Goedel's theorem may be demonstrated using arguments having an information-theoretic flavour. In such an approach, it is possible to argue that if a theorem contains more information than any given set of axioms, then it is impossible for the theorem to have been 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.' 
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Clancy, Susan A, The trauma Myth: The Truth About Sexual Abuse of Chiudren-- and its Aftermath, Basic Books 2010 From Booklist 'Clancy argues the controversial position that survivors of childhood sexual abuse are victimized not only by their abusers, whose acts often leave them more confused (due to incomprehension) than frightened, but also and inadvertently by well-intentioned health professionals, whose interpretations of abusive experiences are often more traumatic than actual events and effects. Well-meaning practitioners emphasize abuse’s violence, fear, and threats, which “do not characterize the experiences that most victims have.” Challenging the traumatogenic model’s assumptions and origins, Clancy questions the “repression” concept of “recovered” memories as oddly selective compared to conceptions of other major traumas. Skillfully interweaving case studies, statistics, and technical data, she disputes that abusive acts destabilize neurobiology as in other traumas. Positing that the trauma model damages victims with inaccurate predictions and ineffective treatments, she recommends tracking consequences via cognitive, behavioural, and developmental pathways because “what hurts most victims is not the experience itself but the meaning of the experience—how victims make sense of what happened . . . how these understandings make them feel about themselves and others.”' --Whitney Scott 
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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' 
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Davis, Martin, The Undecidable : Basic Papers on Problems Propositions Unsolvable Problems and Computable Functions, Raven Press 1965 Description: '[Includes] ... the basic papers of Gödel, Church, Turing, and Post in which the class of recursive functions was singled out and seen to be just the class of functions that can be computed by terminating processes. Also presented is the work of Church, Turing, and Post in which problems from the theory of abstract computing machines, from mathematical logic, and finally from algebra are shown to be unsolvable in the sense that there is no terminating process for dealing with them. Finally, the book presents the work of Kleene and of Post initiating the classification theory of unsolvable problems. Already the standard reference work on the subject, The Undecidable is also ideally suited as a text or supplementary text for courses in logic, philosophy, and foundations of mathematics.'  
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Dawkins, Richard, Climbing Mount Improbable, W. W. Norton & Company 1997 Amazon editorial review: 'How do species evolve? Richard Dawkins, one of the world's most eminent zoologists, likens the process to scaling a huge, Himalaya-size peak, the Mount Improbable of his title. An alpinist does not leap from sea level to the summit; neither does a species utterly change forms overnight, but instead follows a course of "slow, cumulative, one-step-at-a-time, non-random survival of random variants" -- a course that Charles Darwin, Dawkins's great hero, called natural selection. Illustrating his arguments with case studies from the natural world, such as the evolution of the eye and the lung, and the coevolution of certain kinds of figs and wasps, Dawkins provides a vigorous, entertaining defense of key Darwinian ideas.' 
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Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. . . . In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands 
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Feynman, Richard P, and Robert B Leighton et al, The Feynman Lectures on Physics (volume 1) : Mainly Mechanics, Radiation and Heat, Addison Wesley 2011 Foreword: 'This book is based on a course of lectures in introductory physics given by Prof. R P Feynman at the California Institute of Technology during the academic year 1961-62. ... The lectures constitute a major part of a fundamental revision of the introductory course, carried out over a four year period. ... The need for a basic revision arose both from the rapid development of physics in recent decades and from the fact that entering freshmen have shown a stewady incrase in mathematical ability as a result of improvements in high school mathematical course content.' 
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Gödel, Kurt, and Solomon Feferman et al (eds), Kurt Gödel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...' 
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Gödel, Kurt, and B Meltzler (translator), R B Braithwaite (Introduction), On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Dover 1992 A translation of Uber Formal Unentscheidbare Satze der Principia Mathematica und Verwandter Systeme I, Monatshefte fur Mathematik und Physic, 38(1931) 173-198. Jacket: 'In 1931 a young Austrian mathematician published an epoch making paper containing one of the most revolutionary ideas in logic since Aristotle. Kurt Gödel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will mot give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th century mathematics.' 
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Gödel, Kurt, "On formally undecidable propositions of Principia Mathematica and related systems, I" in Solomon Fefferman et al (eds) Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...' 
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Hawking, 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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Streater, Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2000 Amazon product description: 'PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.' 
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von Campenhausen, Hans, and (English translation revised by L A Garrard), The Fathers of the Greek Church, Adam and Charles Black 1963 Professor of Ecclesiastical History, Heidelberg Universityback

Zee (2010), Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2010 ' Since it was first published, Quantum Field Theory in a Nutshell has quickly established itself as the most accessible and comprehensive introduction to this profound and deeply fascinating area of theoretical physics. Now in this fully revised and expanded edition, A. Zee covers the latest advances while providing a solid conceptual foundation for students to build on, making this the most up-to-date and modern textbook on quantum field theory available. This expanded edition features several additional chapters, as well as an entirely new section describing recent developments in quantum field theory such as gravitational waves, the helicity spinor formalism, on-shell gluon scattering, recursion relations for amplitudes with complex momenta, and the hidden connection between Yang-Mills theory and Einstein gravity. Zee also provides added exercises, explanations, and examples, as well as detailed appendices, solutions to selected exercises, and suggestions for further reading.' 
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Papers

Shannon, Claude E, "Communication in the Presence of Noise", Proceedings of the IEEE, 86, 2, February 1998, page 447-457. Reprint of Shannon, Claude E. "Communication in the Presence of Noise." Proceedings of the IEEE, 37 (January 1949) : 10-21. '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 this maximum rate are discussed. The equivalent number of binary digits per second of certain information sources is calculated.' . back

Zurek, Wojciech Hubert, "Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical", Physical Review A, 76, 5, 16 November 2007, page . Abstract: '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 a framework for 'wave-packet collapse', designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment &mdash the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them &mdash into becoming a witness.'. back

Zurek, Wojciech Hubert, "Decoherence, einselection, and the quantum origins of the classical", Review of Modern Physics, 75, , 2003, page 715-775. The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal “Schrödinger-cat states.” The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective “collapse of the wave packet.” The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand “classical reality” as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be “found out” without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their “fitness” in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born’s rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.. back

Links

Alison Crowther & Peter Faulkner, A cave site in Kenya’s forests reveals the oldest human burial in Africa, 'Africa is often referred to as the cradle of humankind – the birthplace of our species, Homo sapiens. There is evidence of the development of early symbolic behaviours such as pigment use and perforated shell ornaments in Africa, but so far most of what we know about the development of complex social behaviours such as burial and mourning has come from Eurasia. However, the remains of a child buried almost 80,000 years ago under an overhang at Panga ya Saidi cave in Kenya is providing important new details. Working with a team of researchers from Kenya, Germany, Spain, France, Australia, Canada, South Africa, the United Kingdom and the United States, we studied the burial. Our results, published in Nature today, reveal valuable insights into human cultural evolution, including how Middle Stone Age populations interacted with the dead.' back

Anshel Pfeffer, Opinion | Does Anyone Really Care About the Haredim? , ' “It is a ruling from heaven, and we cannot know God’s accounts,” the rabbi said. When asked what atonement can be done after the heavenly ruling, he listed a number of useful ideas: “strengthening the study of Torah … women should strengthen their modesty … [more observance] of the commandments of washing the hands before eating, and concentrating on the meaning of the blessings, so to feel the closeness of God.” ' back

Aquinas, Summa I, 28, 1, Are there real relations in God?, 'Reply to Objection 4. Relations which result from the mental operation alone in the objects understood are logical relations only, inasmuch as reason observes them as existing between two objects perceived by the mind. Those relations, however, which follow the operation of the intellect, and which exist between the word intellectually proceeding and the source whence it proceeds, are not logical relations only, but are real relations; inasmuch as the intellect and the reason are real things, and are really related to that which proceeds from them intelligibly; as a corporeal thing is related to that which proceeds from it corporeally. Thus paternity and filiation are real relations in God.' 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

Arrow Leadership, Home - Arrow Leadership, 'Arrow Australia is the premier Christian provider of training, in the non-tertiary sector, for emerging and mature leaders. At Arrow we facilitate development and learning that transforms the lives of tomorrow's leaders today, and shapes how they are able to transform the context of their work/vocation through their ministry and mission. The Arrow Leadership Program was established in Australia in 1995. Since then many groups have progressed though a two-year leadership and spiritual growth program with over 300 graduates.' back

Bayes' theorem - Wikipedia, Bayes' theorem - Wikipedia, the free encyclopedia, 'IIn probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) describes the probability of an event, based on conditions that might be related to the event. . . .Bayes' theorem is named after Rev. Thomas Bayes (1701–1761), who first provided an equation that allows new evidence to update beliefs. It was further developed by Pierre-Simon Laplace, who first published the modern formulation in his 1812 Théorie analytique des probabilités. Sir Harold Jeffreys put Bayes' algorithm and Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry".' back

Bose-Einstein condensate - Wikipedia, Bose-Einstein condensate - Wikipedia, the free encyclopedia, ' A Bose–Einstein condensate (BEC) is a state of matter (also called the fifth state of matter) which is typically formed when a gas of bosons at low densities is cooled to temperatures very close to absolute zero (-273.15 °C). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density, about one-hundred-thousandth (1/100,000) the density of normal air, to ultra-low temperatures. This state was first predicted, generally, in 1924–1925 by Albert Einstein following and crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics.' back

Bubble chamber - Wikipedia, Bubble chamber - Wikipedia, the free encyclopedia, 'A bubble chamber is a vessel filled with a superheated transparent liquid (most often liquid hydrogen) used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser,[1] for which he was awarded the 1960 Nobel Prize in Physics.[2] Anecdotally, Glaser was inspired by the bubbles in a glass of beer; however, in a 2006 talk, he refuted this story, saying that although beer was not the inspiration for the bubble chamber, he did experiments using beer to fill early prototypes.' back

Cantor's theorem - Wikipedia, Cantor's theorem - Wikipedia, the free encyclopedia, ' In mathematical set theory, Cantor's theorem is a fundamental result which 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 simple enumeration of the number of subsets. Counting the empty set as a subset, a set with n elements has a total of n 2>sup> subsets, and the theorem holds because n2>sup> > nfor all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also.' back

Carlo Rovelli, Relational Quantum Mechanics, ' I suggest that the common unease with taking quantum mechanics as a fundamental description of nature (the measurement problem) could derive from the use of an incorrect notion, as the unease with the Lorentz transformations before Einstein derived from the notion of observer-independent time. I suggest that this incorrect notion that generates the unease with quantum mechanics is the notion of observer-independent state of a system, or observer-independent values of physical quantities. I reformulate the problem of the interpretation of quantum mechanics as the problem of deriving the formalism from a set of simple physical postulates. I consider a reformulation of quantum mechanics in terms of information theory. All systems are assumed to be equivalent, there is no observer-observed distinction, and the theory describes only the information that systems have about each other; nevertheless, the theory is complete.' back

Catherine Wong, China sends ‘warning shot to Australia’ as patience wears thin, 'China’s latest squeeze on Australia may be largely symbolic, meant as a warning shot for Canberra and other middle powers in the age of US-China rivalry, observers said. In the latest in a series of moves, China’s National Development and Reform Commission said on Thursday it would “indefinitely suspend” all activities under the China-Australia Strategic Economic Dialogue, a forum launched in 2014 and last convened in 2017. The economic planning agency said it made the decision because Canberra’s “Cold War mindset and ideological discrimination” had disrupted cooperation. back

CERN, LHC Homepage, 'The Large Hadron Collider (LHC) sits in a circular tunnel 27 km in circumference. The tunnel is buried around 50 to 175 m. underground. It straddles the Swiss and French borders on the outskirts of Geneva. The first collisions at an energy of 3.5 TeV per beam took place on 30th March 2010. The LHC is designed to collide two counter rotating beams of protons or heavy ions. Proton-proton collisions are foreseen at an energy of 7 TeV per beam. The beams move around the LHC ring inside a continuous vacuum guided by magnets. The magnets are superconducting and are cooled by a huge cryogenics system. The cables conduct current without resistance in their superconducting state. The beams will be stored at high energy for hours. During this time collisions take place inside the four main LHC experiments.' back

Codec - Wikipedia, Codec - Wikipedia, the free encyclopedia, 'A codec is a device or computer program capable of encoding or decoding a digital data stream or signal. Codec is a portmanteau of coder-decoder or, less commonly, compressor-decompressor.' back

Complemented lattice - Wikipedia, Complemented lattice - Wikipedia, the free encyclopedia, ' In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0. Complements need not be unique. A relatively complemented lattice is a lattice such that every interval [c, d], viewed as a bounded lattice in its own right, is a complemented lattice. An orthocomplementation on a complemented lattice is an involution which is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law is called an orthomodular lattice.' back

Convex set - Wikipedia, Convex set - Wikipedia, the free encyclopedia, 'In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.' back

Conway's Game of Life - Wkipedia, Conway's Game of Life - Wikipedia, the free encyclopedia, ' The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970.[1] It is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further input. One interacts with the Game of Life by creating an initial configuration and observing how it evolves. It is Turing complete and can simulate a universal constructor or any other Turing machine.' back

Daniel W Graham - Heraclitus, Heraclitus, 'A Greek philosopher of Ephesus (near modern Kuşadası, Turkey) who was active around 500 BCE, Heraclitus propounded a distinctive theory which he expressed in oracular language. He is best known for his doctrines that things are constantly changing (universal flux), that opposites coincide (unity of opposites), and that fire is the basic material of the world. The exact interpretation of these doctrines is controversial, as is the inference often drawn from this theory that in the world as Heraclitus conceives it contradictory propositions must be true.' back

De Gaulle (film) - Wikipedia, De Gaulle (film) - Wikipedia, the free encyclopedia, ' En mai 1940, Charles de Gaulle, fraichement nommé général de brigade, est confronté à l’effondrement militaire et politique de la France. Il s'oppose alors au défaitisme du gouvernement en place, en partie incarné par le maréchal Philippe Pétain. Après avoir fui à Bordeaux avec certains membres du gouvernement, Charles de Gaulle rejoint Londres pour demander l'aide de Winston Churchill et tenter de maintenir la lutte. De son côté, sa femme Yvonne doit quitter la propriété de Colombey les Deux Églises. Elle se retrouve avec ses trois enfants en plein exil et doit échapper à l’arrivée des Allemands. Elle part d'abord pour le Loiret, puis à Carantec en Bretagne, avant de fuir le pays en bateau. La vie est dure notamment pour la benjamine de la famille, Anne, atteinte de trisomie 21. Depuis Londres, le général va tout tenter pour remotiver les troupes. Le 18 juin 1940, il lance un appel radio sur les ondes de la BBC1,2. Le lendemain, il retrouve sa famille.' back

Distribution (mathematics) - Wikpedia, Distribution (mathematics) - Wikpedia, the free encyclopedia, ' Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.' back

Easter Island - Wikipedia, Easter Island - Wikipedia, the free encclopedia, 'Easter Island (Rapa Nui: Rapa Nui, Spanish: Isla de Pascua) is a Polynesian island in the southeastern Pacific Ocean, at the southeasternmost point of the Polynesian Triangle. Easter Island is famous for its 887 extant monumental statues, called moai, created by the early Rapa Nui people. In 1995, UNESCO named Easter Island a World Heritage Site, with much of the island protected within Rapa Nui National Park.' back

Editors: Encyclopaedia Britannica, Pare-mutuel, 'Pari-mutuel, ( French: pari, “bet”; mutuel, “mutual”) plural pari-mutuels, or Paris-mutuels, method of wagering introduced in France about 1870 by Parisian businessman Pierre Oller. It became one of the world’s most popular methods of betting on horse races.' back

Einstein, Podolsky and Rosen, Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' back

Erlangen program - Wikipedia, Erlangen program - Wikipedia, the free encyclopedia, 'An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. This Erlangen Program (Erlanger Programm) — Klein was then at Erlangen — proposed a new solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory.' back

Hexapla - Wikipedia, Hexapla - Wikipedia, the free encyclopedia, 'Hexapla (Ἑξαπλά: Gr. for "sixfold") is the term for an edition of the Bible in six versions. Especially it applies to the edition of the Old Testament compiled by Origen of Alexandria, which placed side by side: Hebrew Hebrew transliterated into Greek characters Aquila of Sinope Symmachus the Ebionite A recension of the Septuagint, with (1) interpolations to indicate where the Hebrew is not represented in the Septuagint -- these are taken mainly from Theodotion's text and marked with asterisks, and (2) indications, using signs called obeloi (singular: obelus), of where words, phrases, or occasionally larger sections in the Septuagint do not reflect any underlying Hebrew. Theodotion.' back

Hoi-Kwan Lau & Aashish A. Clerk, Macroscale entanglement and measurement, back

Holy See, Congregation for the Doctrine of the Faith, Profile: 'CONGREGATION FOR THE DOCTRINE OF THE FAITH Founded in 1542 by Pope Paul III with the Constitution "Licet ab initio," . . . The only curial organism which is older is the Secretariat of State, whose forerunner, the Apostolic Secretariat, was created by Innocent VIII on December 31, 1487, with the Constitution "Non debet reprehensibile." . . . Today, according to Article 48 of the Apostolic Constitution on the Roman Curia, 'Pastor Bonus,' promulgated by the Holy Father John Paul II on June 28, 1988, "the duty proper to the Congregation for the Doctrine of the Faith is to promote and safeguard the doctrine on the faith and morals throughout the Catholic world: for this reason everything which in any way touches such matter falls within its competence." The congregation is now headed by Prefect Cardinal Joseph Ratzinger. It has a secretary, Archbishop Tarcisio Bertone, S.D.B., named June 13, an under-secretary, Msgr. Jozef Zlatnansky, and a staff of 32, according to the 1995 "Annuario Pontificio" or "Pontifical Yearbook." It also has 23 members - cardinals, archbishops and bishops - and 27 consultors. Given the nature of its task, congregation work is divided into four distinct sections: the doctrinal office, the disciplinary office, the matrimonial office and that for priests. back

Homeomorphism - Wikipedia, Homeomorphism - Wikipedia, the free encyclopedia, 'In the mathematical field of topology, a homeomorphism or topological isomorphism (from the Greek words (homoios) = similar and (morph) = shape = form (. . . ) is a bicontinuous function between two topological spaces. Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings which preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.' back

Hugh Breakey, Friday essay: 3 ways philosophy can help us understand love, ' Love can seem a primal force, an intoxicating mix of desire, care, ecstasy and jealousy hard-wired into our hearts. The polar opposite of philosophy’s measured rationality and theoretical speculations. Yet if you take any topic in the world, and keeping asking deep questions of it, you will ultimately wind up doing philosophy. Love is no different. Indeed, many famous philosophers— Kant, Aristotle, De Bouvier — wrote about love and how it fitted into their larger theories of human reason, excellence and freedom. Unsurprisingly, their historically-situated views tended to mirror the culturally valued types of love in their time. The Greeks eulogised the love of friendship. Scholars in the middle ages ruminated on the love of God. With the Renaissance, romantic love moved centre stage.' back

Hypatia - Wikipedia, Hypatia - Wikipedia, the free encyclopedia, 'Hypatia (Greek: Ὑπατίᾱ Hupatíā; born c. 350–70; died 415),often called Hypatia of Alexandria (Ancient Greek: Ὑπατίᾱ η Αλεξανδρινή), was a Greek mathematician, astronomer, and philosopher in Egypt, then a part of the Byzantine Empire. She was the head of the Neoplatonic school at Alexandria, where she taught philosophy and astronomy. According to contemporary sources, Hypatia was murdered by a Christian mob or by Christian zealots known as Parabalani after being accused of exacerbating a conflict between two prominent figures in Alexandria, the governor, Orestes, and the bishop, Cyril of Alexandria.' back

IUPAC nomenclature - Wikipedia, IUPAC nomenclature - Wikipedia, the free encyclopedia, 'IUPAC nomenclature is a system of naming chemical compounds and of describing the science of chemistry in general. It is developed and kept up to date under the auspices of the International Union of Pure and Applied Chemistry (IUPAC). ...

This article treats the system of nomenclature in general, notably its aims and historical development. Separate articles treat the naming of organic compounds and inorganic compounds in more detail.' . . .

The primary function of chemical nomenclature is to ensure that the person who hears or reads a chemical name is under no ambiguity as to which chemical compound it refers to: each name should refer to a single substance. It is considered less important to ensure that each substance should have a single name, although the number of acceptable names is limited.' back

Jeffrey Nicholls (2019), A prolegomenon to scientific theology, ' This thesis is an attempt to carry speculative theology beyond the apogee it reached in the medieval work of Thomas Aquinas into the world of empirical science. Since the time of Aquinas, our understanding of the Universe has increased enormously. The ancient theologians not only conceived a perfect God, but they also saw the world as a very imperfect place. Their reaction was to place God outside the world. I will argue that we live in a Universe which approaches infinity in size and complexity, is as perfect as can be, and fulfils all the roles traditionally attributed to God, creator, lawmaker and judge.' back

Joanna Ossinger, It’s Hard to Take Dogecoin Seriously, But the Doge Doesn’t Care, ' There’s no good reason it should be valuable. “Dogecoin has no apparent commercial or investment use other than as a conduit for speculative mania and the attempt to make a buck,” says Jeffrey Halley, senior market analyst at Oanda Asia Pacific Pte., a platform for trading traditional currencies. “I suspect much of its appeal lies in the fact that it is very, very cheap to buy and sell, as opposed to $60,000 for Bitcoin, making it much more approachable to a retail trader who fancies a flutter.” ' back

John Chrysostom - Wikipedia, John Chrysostom - Wikipedia, the free encyclopedia, 'John Chrysostom (c. 347–407, Greek: Ἰωάννης ὁ Χρυσόστομος), Archbishop of Constantinople, was an important Early Church Father. He is known for his eloquence in preaching and public speaking, his denunciation of abuse of authority by both ecclesiastical and political leaders, the Divine Liturgy of St. John Chrysostom, and his ascetic sensibilities. After his death (or, according to some sources, during his life) he was given the Greek surname chrysostomos, meaning "golden mouthed", rendered in English as Chrysostom.' 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

Max Fisher, ‘Belonging Is Stronger Than Facts’: The Age of Misinformation, ' Social and psychological forces are combining to make the sharing and believing of misinformation an endemic problem with no easy solution. . . . We are in an era of endemic misinformation — and outright disinformation. Plenty of bad actors are helping the trend along. But the real drivers, some experts believe, are social and psychological forces that make people prone to sharing and believing misinformation in the first place. And those forces are on the rise.' back

Max Hastings, The Vietnam War’s Lessons Went Unlearned in Afghanistan, ' The first Vietnam message is that the Northerners won that war not because they were communists, but mostly because they were Vietnamese. Almost everybody hates being bossed around by foreigners. Throughout the war years, Vietnamese knew that no representative of the Saigon government could leave his bed in the morning without asking his American paymasters which side to get out. A Southerner named Chau Phat said: “The communists could ceaselessly remind us how humiliating it was to be occupied.” ' back

No cloning theorem - Wikipedia, No cloning theorem - Wikipedia, the free encyclopedia, 'The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields.' back

Projection (linear algebra) - Wikipedia, Projection (linear algebra) - Wikipedia, the free encyclopedia, ' In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection.' back

Richard Stone, ‘It’s like the embers in a barbecue pit.’ Nuclear reactions are smoldering again at Chernobyl, ' Thirty-five years after the Chernobyl Nuclear Power Plant in Ukraine exploded in the world’s worst nuclear accident, fission reactions are smoldering again in uranium fuel masses buried deep inside a mangled reactor hall. “It’s like the embers in a barbecue pit,” says Neil Hyatt, a nuclear materials chemist at the University of Sheffield. Now, Ukrainian scientists are scrambling to determine whether the reactions will wink out on their own—or require extraordinary interventions to avert another accident.' back

Sasha Borissenko, Why New Zealand’s foreign minister is her own woman, ' “As a Māori woman there is an embedded sense of social justice, and striving for equality of opportunity and Indigenous advancement,” she told Al Jazeera. “If you’ve been brought up in a Māori community you will have experienced some form of mistreatment and at an extreme level – racism.” The daughter of Sir Robert Te Kotahitanga – the adopted son of Māori King Koroki – Mahuta grew up assisting her father in key treaty negotiations. In 2016, Mahuta became the first woman to display a moko kauae (sacred facial tattoo) in Parliament, and last year chalked up another first – becoming New Zealand’s first female foreign minister.' back

Special linear group - Wikipedia, Special linear group - Wikipedia, the free encyclopedia, 'In mathematics, the special linear group of degree nM/i< over a field F is the set of n × nM/i> matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.' back

Steven Johnson, How Humanity Gave Itself an Extra Life, ' The descendants of English and Welsh babies born in 1918, who on average lived just 41 years, today enjoy life expectancies in the 80s. And while Western nations surged far ahead in average life span during the first half of the last century, other nations have caught up in recent decades, with China and India having recorded what almost certainly rank as the fastest gains of any society in history. A hundred years ago, an impoverished resident of Bombay or Delhi would beat the odds simply by surviving into his or her late 20s. Today average life expectancy in India is roughly 70 years.'
Steven Johnson’s article is excerpted from his 13th book, “Extra Life: A Short History of Living Longer.” back

Synesius - Wikipedia, Synesius - Wikipedia, the free encyclopedia, 'Synesius (Greek: Συνέσιος; c. 373 - c. 414), a Greek bishop of Ptolemais in the Libyan Pentapolis after 410, was born of wealthy parents, who claimed descent from Spartan kings, at Cyrene between 370 and 375.' back

Wang Xiangwei, China’s treatment of Oscar winner Chloé Zhao shows how to lose friends and not influence people, ' The contrast could not be more telling. While the international media hailed Chloé Zhao as making history this week by becoming the second woman and the first Asian woman to land the best director Oscar award for the film Nomadland, the Chinese media acted as if her historic win had not happened at all. Moreover, international attention has since then focused on the Chinese censors’ heavy-handed approach in scrubbing any mention of the award from social media and search engines because of her critical comments about the country of her birth in a 2013 interview. China has missed a great opportunity to promote its soft power by refusing to celebrate her win and hold her up as an inspiration to women everywhere.' back

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