natural theology

We have just published a new book that summarizes the ideas of this site. Free at Scientific Theology, or, if you wish to support this project, buy at Scientific Theology: A New Vision of God

Contact us: Click to email

Notes

Sunday 30 August 2020 - Saturday 5 September 2020

[Notebook: DB 85 Science]

[page 143]

Sunday 30 August 2020

Oliver Passon: Bohmian Mechanics Oliver Passon: What you always wanted to know about Bohmian mechanics but were afraid to ask

Passon page 3: Bohmian mechanics was first developed by Louis de Broglie = "pilot wave theory" → Bohm 1952.

Peter Holland: Notes 2017 3 September. Read Holland in 1993-4 Notebook WFA (5), (47) (6, Land and Environment Court Transcript) Peter Holland: The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics

John Bell: de Broglie-Bohm theory → Bell inequality; Birbeck College group: "ontological" or "causal" interpretation of quantum mechanics. John Bell: Speakable and Unspeakable in Quantum Mechanics

page 4: Physical system described not by wave function alone but by wave function ψ + configuration Qi: ψ → (ψ, Qi) = qm → de Broglie-Bohm. de BB theory supplements quantum mechanics with particle positions. Particles are guided by the ψ field.

page 7: deBB theory is deterministic since ψ + Qi fix evolution uniquely. So no creation. Randomness arises from averaging over ignorance.

[page 144]

No need for complementarity since wave and particle both have a role.

Non-local since ψ is defined on the configuration space Qi → Bell's inequality.

Passon page 8: Measurement has no special role because the system is deterministic

'Within ordinary quantum mechanics the identification of "observables" with linear Hilbert space operators is usually regarded as the key innovation. Their non-commutativity is believed to be the mathematical embodiment of the deep epistemological lesson quantum mechanics teaches us.'

Energy, angular momentum, spin etc have different ontological status from position, Qi .

page 9: These properties are called "contextual" depending on experimental arrangement. Bohmian particles have no properties other than position.' So why have different particles with different properties?

page 10: Solution to measurement problem:

a) wave function is complete
b) wave function evolves unitarily and deterministically
c) meaurements have deterministic outcomes
but in a two or more state system the measurement leads to a mixed state yielding either a, b, c, . . . so c) contradicts a) and b).' All we are doing here is applying requisite variety so initial states of low entropy cannot determine the final state of high entropy. As von Neumann notes, measurement is creative [increases entropy]. John von Neumann: The Mathematical Foundations of Quantum Mechanics Chapter 5, pp 227 sqq.

[page 145]

Passon page 11: Quantum mechanics fails to distinguish actual outcome of measurement, whereas deBB, with the help of the Qi does.

page 12: 'most common objection to deBB is based on non-locality and apparent conflict with relativity and quantum field theory. Can we make deBB i) reproduce QFT; and ii) have non-relativistic limiting case?

Bell 1986 page 174: ' "In particular we exclude the notion of 'observable' in favour of that of 'beable'. The beables of a theory are those elements which might correspond to elements of reality, to things which exist. . . . Indeed observation and observers must be made of beables." ' ie classical entities [which they are?].

page 14; Field theoretical generalization of deBB
QFT generalizations distinguished by definitions of beables:
[type] i. Field beables for bosons, particle beables for fermions
[type] ii. Bosons field beables, fermions not beables

page 15: [type] iii. ? Bell: Fermions are beables

page 17: Remarks on theory generalization:

page 18; Philosophy of science - Intertheory relations. We may say T1 "reduces to" T2, or its inverse, T1 is a generalization of T2.

Nagel 1961: T2 is a reduction that can be deduced from T1. - 'reduction' means move from general to specific. T2 might be a limiting case of T1, eg clasical mechanics comes from general relativity when v/c2 → 0, and ℏ → 0

page 20: deBB gives observer-independent account of quantum phenomena - rubbish.

[page 146]

All this waffle makes me a lot more confident about my approach which devalues determinism so allowing for creation by some more sophisticated version of the big bang.

Monday 31 August 2020
Tuesday 1 September 2020

Tess Tess (1979 film) - Wikipedia

Principia theologica is starting to work after 2 months, a retitling of e30_cognitive_cosmology, which will be the new chapter 6 of scientific-theology.com, Constructing the Divine World. Slowly becoming convinced that writing this stuff is a real job and trying to keep up to trade standards producing something useful, reliable, durable, sustainable, green and above all peaceful — fundamentally still working on the theory of peace that started in the 80s. I am over the disappointment of my honours year now and feeling confident (as I did at the beginning of that year) that I can go on to do a PhD (or equivalent) to get my ideas into the academic stream. naturaltheology.net/History/ATheoryOfPeace1987

Wednesday 2 September 2020

How can I bring out the music of the divine world, built into Hilbert space? [page 147]

Ones feelings of authenticity are strongly influenced by ones perceptions of what other people and institutions think of one. Fate has put me between two major forces, the Roman Catholic theological establishment and the high energy nuclear physics establishment, both of which are quite likely to be so confident in their positions of power that they can either completely ignore me or laugh me out of court. My only real recourse is to be right and stand by my position, which means that every point I make must be soundly based. This I try to do, but nevertheless I do feel a bit like a charlatan and can see why many people are deeply distrustful of allegedly scientific and evidence based positions that go against established views. My task remains to make myself asymptotic to the truth which is of course evolving in my own mind so I am a bit doubtful of my story and my only hope is to complete the circuit of change and reach closure, that is consistency, the event that so excited Einstein when he finally completed the general theory. What I must do then is examine every detail of my story to see that it is both well grounded in the available evidence and that my whole story is consistent. The beauty of Einstein's work is that it culminated in a relatively simple field equation backed by a deep vein of tacit mathematics which has been shown to explain the observed phenomena of the universe perfectly. How do I get to this level of perfection? My 'field equation' is in effect the quantum of action and the tacit backing of this equation is the set of computable functions. The new bit that has become clear in the last few weeks it that the linear operators in a countably infinite Hilbert space are turing computable so that everything we observe is locally consistent.

Thursday 3 September 2020

[page 149]

Principia: Classical physics founded on differentiable manifold and tensors. Quantum physics founded on Hilbert space.

Then cybernetics - communication and control are central issues in theology, ie revelation from god and divine providence. Central to the design of military automats. This is an exciting characterization and takes us back to the dawn of weapons design, eg the woomera. Woomera (spear-thrower) - Wikipedia

Classical physics invented the variational principe via the lagrangian which takes a holistic view of processes which involve a quantum of action, equivalent to nh where 0 ≥ n0.

Einstein on Planck: 'I have often heard colleagues try to attribute this attitude of [Planck's] to his extraordinary willpower and self discipline - wrongly in my opinion. The state of mind which enables a man to do work of this kind is akin to that of the religious worshipper or lover; the daily effort comes from no deliberate intention or program, but straight from the heart. . . . May he succeed in uniting the quantum theory and electrodynamics in a single logical system.' Albert Einstein: Einstein's Essays in Science, page 5

Three minutes on the loo reveals an interesting insight. The easiest things to understand and model are linear. Classical physics linearises space-time by infinitesimalizing it because second order effects can be considered irrelevant at the infinitesimal level of resolution. Quantum theory, on the other hand is inherently linear insofar as all its operators are linear and unitary. It is linear because it is the fundamental and simple theory of the universe using what we might call naturally linear operators instead of using infinitesimals to

[page 149]

obtain linerarity. The natural 'infinitesimal' in quantum mechanics is the quantum of action.

Riemann on differential equations as the foundation of science [Brown section 9.2 Up to a Diffeomorphism] ' "As is well known, physics became a science only after the invention of differential calculus . . . True basic laws hold in the small, and must be formalized as differential equations." ' Kevin Brown: Reflections on Relativity, page 628

Einstein page 18: ' The physical world is represented as a four-dimensional continuum. If I assume a Riemannian metric in it and ask what are the simplest laws that such a metric can satisfy, I arrive at the relativist theory of gravitation in empty space.'

page 19: ' The important point for us to observe is that all these constructions and laws connecting them can be arrived at by the principle of looking for the simplest concepts that link them.'

page 27: ' It seems that the human mind must first construct forms independently before we can find them in things.'

page 29; ' Actual results to support the belief in a complete chain of physical causation hardly existed before Newton.

page 30: re Riemann: ' The differential law is the only form which completely

[page 150]

satisfies the modern physical demand for causality.' I would say wrong. We need logical continuity, overlapping ends like Aristotle's middle term. Aristotle: Physics V, iii, 227a10 sqq

Einstein page 31: ' It was only by considering what takes place during an infinitely short time (the differential law) that Newton reached a formula that applied to all motion whatever.'

' The connection of force and acceleration was only made possible for him by the introduction of the concept of mass.'

page 32: ' The only things that figure as the causes of the acceleration of the masses of a system are the masses themselves.' MASS ≡ CHARGE

page 34: [Newton] is aware that space must possess a kind of physical reality if the laws of motion are to have any meaning, a reality of the same sort as the material points and the intervals between them.'

page 38: ' The application of Newtonian mechanics to continuously distributed masses led inevitably to the discovery of partial differential equations which in their turn first provided the language for the laws of field theory.'

page 39: "The de Broglie - Schrödinger method . . . does indeed deduce the existence of only-discrete states in astonishing agrement with empirical fact, on the basis of differential equations operating with a kind of resonance theory . . . "

page 43: ' In a particular department of theoretical physics the continuous

[page 151]

field thus appeared side by side with the material points . . . This dualism remains even today, disturbing as it must be to every orderly mind.' So we replace the duality with the unity of action and logic.

Einstein page 44: 'After Maxwell [people] conceived physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations.'

page 45: '. . . the quantitites which figure in [quantum mechanic's] laws make no claim to decribe physical reality itself, but only the probabilities of a physical reality which we have in view.' which corresponds to some model we have constructed of (eg) an electron or a photon [which defines a "bin" ( event) whose contents we count to verify the predicted probability].

page 48 Relativity: 49: Time is local, breaking through from the initial moment.

page 50: 'the general theory of relativity, which based on the equality of inertia and weight, provides a theory of gravitational field.'

page 51: Generalization of metric . . . had already been accomplished in the sphere of mathematics through the researches of Gauss and Riemann. Albert Einstein: The Problem of Space, Ether and Field in Physics

64: 'All spatial relations are reduced to those of contact (the intersections of of straight lines and planes, points lying on straight lines, etc). Space as a continuum does not figure in [Euclid's] conceptual scheme at all.'

page 65: ' In the Cartesian treatment . . . all surfaces are in principle equally represented without any arbitrary preference for linear figures in the construction of geometry.'

[page 152]

Einstein page 66: ' Newton's acceleration is only thinkable or definable in relation to space as a whole.'

page 68: 'Physical space and the ether are only different names for the same thing,'

'Only the genius of Riemann had . . . won its way . . . to a new conception of space in which space was deprived of its rigidity and in which its power to take part in physical events was recognized as possible.'

'With the discovery of the relativity of simultaneity, space and time were merged into s single continuum . . . ' because in the layered model of the emergence of the universe they are manifestations of a single quantum source, energy-momentum.

page 69: ' . . . the observed fact is undoubtedly the supreme arbiter; but it cannot pronounce sentence until the wide chasm separating the axioms from their verifiable consequences has been bridged . . . '

page 70: '. . . it is the line of thought that led from the special to the general theory of relativity and thence to its latest offshoot, the unitary field theory.' [meaning, I think 'unified' rather than 'unitary' as in quantum theory, although the algebraic meaning of unitary may be appropriate]

page 71; ' the defining equation of the metric . . . is nothing but the Pythagorean theorem applied to the differentials of the coordinates.'

' Only those equations are admissible as expressions of natural law

[page 153]

which do not change their form when the coordinates are changed by Lorentz transformation (covariance of equations in relation to Lorentz transformations). This method led to the discovery of the necessary connection between impulse [momentum] and energy, the strength of an elecric and a magnetic field, electrostatic and electromagnetic forces, inert mass and energy; and the number of independent concepts and fundamental equations was thereby reduced.

Einstein page 72: 'If we elevate the equivalence of all coordinate system for the formulation of natural laws into a principle, we arrive at the general theory of relativity . . . ' [insofar as addressed networks are their own coordinate systems, all this process is unnecessary, all we need is a system [domain name server] to handle mappings between discrete addresses].

page 73: 'The atructure of such space differs absolutely in one respect from that of Euclidean space. The coefficients gμν [in ds2 = gμνdxμdxν] are for the time being any functions whatever of the coordinates x1 to x4 and the structure of space is not really determined until these functions gμν are known.

' The problem of gravitation is thus reduced to a mathematical problem: it was required to find the simplest fundamental equations which are invariant in relation to any coordinate transformation whatever.'

page 75: Since we are looking for a space which exceeds Riemannian space in wealth of structure, the obvious thing is to enrich Riemannian space by adding the relation of direction of parallelism.' More to the point, we want to derive the Riemannian space from the Hilbert space by general covariance.

[page 154]

Friday 4 September

Einstein page 109: 'No space and no portion of space is without gravitational potentials; for these give it the metrical properties without which it is not thinkable at all. The existence of the gravitational field is directly bound up with the existence of space.'

So how does the space-time metric relate to the metric in Hilbert space? How is potential created - it is something to do with the emergence of time and energy from pure action [or simply the innate tendency of pure action, like god, to create itself].

page 110: ' In the present state of theory it looks as if the electromagnetic field as compared with the gravitational field were based on a completely new formal motive.'

All the vectors in Hilbert space have phase which we can consider to be equivalent to time, so time is in effect built in to Hilbert space, so we have here a first step toward identifying space-time and Hilbert space. The second step relies on the velocity of light which creates a bond between time and space, so that frequency in time is mapped to the frequency of plane waves in space, but where does the velocity of light come from? We may assume that pure quantum mechanics exists prior to space-time and its character is reflected in spacetime by the existence of null geodesic, that is situations where entities travellng at the velocity of light like photons are in effect outside space-time, that is representatives of the state 'before' space-time.

[page 155]

emerged. To create spacetime as we know it we need to create states that travel at less than the velocity of light, that is massive states. So, in the quotes from Einstein above (Einstein page 34). So the creation of space-time seems to be connected to he creation of photons and massive particles (electrons).

We imagine that the first space-time to be created cannot be inertial because it is not closed, whereas the initial singularity which continues to exist as the primordial hardware of the universe is a closed system in the sense that there is nothing outside it so it contains its own boundary, which from a logical point of view is the region of local inconsistency which does not exist, so we can say that the boundary of the initial singularity (and the universe that emerges within it is the non-existence which lies outside existence. Given a boundary, we might guess that the other conditions of fixed point theory are fulfilled, so the system will have a tendency to form fixed points by mapping onto itself and we might identify these as the bubbles of energy we call particles which are in effect images of the initial singularity in a way analogous to the fact that we (and every other particle in the universe) are images of god.

We imagine that because the system is uncontrolled it tries everything and only those systems which are locally consistent continue to exist, and this process gives birth to Hilbert space [starting with zero or 1 dimension] and then the space-time of general relativity [emerged] and becomes 4D by the need for time division multiplexing and no crossed wires to give rise to [orthogonal regions of] existence within the region of local annihilation.

[page 156]

Since we need massive particles to create space-time we might guess that electrons, positrons and photons preceded gravitation and that gravitational potential is a very weak by-product of mostly neutralized electrical potential in a manner analogous to the binding of nucleons in a nucleus is a by-product of the strong force acting within the nuclei. We see all this accompanied by variation and selection creating enough basic opcodes (particles) to enable the formation of a universal computer network. This is quite a good yarn, and we should tailor the principia theologica to make it sound plausible enough to sell. The fundamental principle remains: pure action will try everything, local consistency will allow only the fit to survive [surrounded by a cloud of temporarily fit particles that may only exist for zillionths of a second], and we develop the little ecosystem of fit particles that lies at the foundation of the universe.

Mathematics is a subset of human language and we find very few if any mathematical books and papers which are constructed purely of mathematical language not embedded in some natural language explanation. At a very rough guess, looking at the encyclopedia of mathematics, we might say that the ratio of natural language symbols to mathematical language symbols is about twenty to one. So the theory of the universe, ie the theology, that I am submitting here [principia theologica] is written almost entirely in natural language to serve as a shell in which a mathematical version of the theology can be generated using appropriate bits and pieces of standard theoretical physics to fill in the details. Michiel Hazewinkel: Encyclopaedia of Mathematics (6 vols)

Saturday 5 September 2020

Hilbert space is a space of orthogonal actions represented by

[page 157]

dynamic complex vectors which have sufficient variety to map onto turing machines. Hilbert space - Wikipedia

Copyright:

You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.

Further reading

Books

Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality . . . . This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality. 
Amazon
  back

Brown, Kevin, Reflections on Relativity, lulu com 2018 ' "Reflections on Relativity" is a comprehensive presentation of the classical, special, and general theories of relativity, including in-depth historical perspectives, showing how the relativity principle has repeatedly inspired advances in our understanding of the physical world.' 
Amazon
  back

Christie, Agatha, A Carribean Mystery, Collins 1964 back

Coulson, Charles A, Valence, Oxford University Press 1985  
Amazon
  back

Dawkins, Richard, The God Delusion, Houghton Mifflin 2006 Amazon Editorial Review From Publishers Weekly 'The antireligion wars started by Daniel Dennett and Sam Harris will heat up even more with this salvo from celebrated Oxford biologist Dawkins. For a scientist who criticizes religion for its intolerance, Dawkins has written a surprisingly intolerant book, full of scorn for religion and those who believe. But Dawkins, who gave us the selfish gene, anticipates this criticism. He says it's the scientist and humanist in him that makes him hostile to religions—fundamentalist Christianity and Islam come in for the most opprobrium—that close people's minds to scientific truth, oppress women and abuse children psychologically with the notion of eternal damnation. While Dawkins can be witty, even confirmed atheists who agree with his advocacy of science and vigorous rationalism may have trouble stomaching some of the rhetoric: the biblical Yahweh is "psychotic," Aquinas's proofs of God's existence are "fatuous" and religion generally is "nonsense." The most effective chapters are those in which Dawkins calms down, for instance, drawing on evolution to disprove the ideas behind intelligent design. In other chapters, he attempts to construct a scientific scaffolding for atheism, such as using evolution again to rebut the notion that without God there can be no morality. He insists that religion is a divisive and oppressive force, but he is less convincing in arguing that the world would be better and more peaceful without it.' Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. 
Amazon
  back

Einstein, Albert, and Translated by Alan Harris , Einstein's Essays in Science, Philosophical Library / Dover 1934, 2009 'His name is synonymous with "genius," but these essays by the renowned physicist and scholar are accessible to any reader. In addition to outlining the core of relativity theory in everyday language, Albert Einstein presents fascinating discussions of other scientific fields to which he made significant contributions. The Nobel Laureate also profiles some of history's most influential physicists, upon whose studies his own work was based. Assembled during Einstein's lifetime from his speeches and essays, this book marks the first presentation to the wider world of the scientist's accomplishments in the field of abstract physics. Along with relativity theory, these articles examine the methods of theoretical physics, principles of research, and the concept of scientific truth. Einstein's speeches to audiences at Columbia University and the Prussian Academy of Science appear here, along with his insightful observations on such giants of science as Johannes Kepler, Sir Isaac Newton, James Clerk Maxwell, Niels Bohr, Max Planck, and others.' 
Amazon
  back

Hazewinkel, Michiel, Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' 
Amazon
  back

Holland, Peter R, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press 1993 Jacket: 'This book presents the first comprehensive exposition of the interpretation of quantum mechanics pioneered by Louis de Broglie and David Bohm. . . . Developing the theme that a material system such as an electron is guided by a surrounding quantum wave, a detailed examination of the classic phenomena of quantum theory is presented . . . The theory provides a novel and satisfactory framework for analysing the classical limit of quantum mechanics and Heisenberg's relations, and implies a theory of measurement without wavefunction collapse. It also suggests a strikingly novel view of relativistic quantum theory, including the Dirac equation, quantum field theory and the wavefunction of the universe.' 
Amazon
  back

Jones, Alexander (ed), The Jerusalem Bible, Darton Longman and Todd 1966 Editor's Foreword: '. . . The Bible . . . is of its nature a written charter guaranteed (as Christians believe) by the Spirit of God, crystallised in antiquity, never to be changed . . . . This present volume is the English equivalent of [La Bible de Jerusalem] . . . an entirely faithful version of the ancient texts which, in doubtful points, preserves the text established and (for the most part) the interpretation adopted by the French scholars in the light of the most recent researches in the fields of history, archaeology and literary criticism.' (v-vi) 
Amazon
  back

Kauffman, Stuart, At Home in the Universe: The Search for Laws of Complexity, Oxford University Press 1995 Preface: 'As I will argue in this book, natural selection is important, but it has not laboured alone to craft the fine architectures of the biosphere . . . The order of the biological world, I have come to believe . . . arises naturally and spontaneously because of the principles of self organisation - laws of complexity that we are just beginning to uncover and understand.'  
Amazon
  back

Klein, Richard G, The Human Career: Human Biological and Cultural Origins, University of Chicago Press 2009 ' Since its publication in 1989, The Human Career has proved to be an indispensable tool in teaching human origins. This substantially revised third edition retains Richard G. Klein’s innovative approach while showing how cumulative discoveries and analyses over the past ten years have significantly refined our knowledge of human evolution. . . . In addition to outlining the broad pattern of human evolution, The Human Career details the kinds of data that support it. For the third edition, Klein has added numerous tables and a fresh citation system designed to enhance readability, especially for students. He has also included more than fifty new illustrations to help lay readers grasp the fossils, artifacts, and other discoveries on which specialists rely. With abundant references and hundreds of images, charts, and diagrams, this new edition is unparalleled in its usefulness for teaching human evolution.' 
Amazon
  back

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

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

Polanyi, Michael, and Amaryta Sen (foreword), The Tacit Dimension, University Of Chicago Press 1966, 2009 Amazon product description: '“I shall reconsider human knowledge by starting from the fact that we can know more than we can tell,” writes Michael Polanyi, whose work paved the way for the likes of Thomas Kuhn and Karl Popper. The Tacit Dimension argues that tacit knowledge—tradition, inherited practices, implied values, and prejudgments—is a crucial part of scientific knowledge. Back in print for a new generation of students and scholars, this volume challenges the assumption that skepticism, rather than established belief, lies at the heart of scientific discovery.' 
Amazon
  back

Suzuki, Daisetz Teitaro, Studies in Zen, Rider and Co, for the Buddhist Society 1953 Studies in Zen is the eigth volume of the collected works of DT Suzuki. Jacket: 'These studies, packed with the jewels of Zen wisdom, and written with unrivalled knowledge, will appeal to all who seek a deeper understanding of Eastern ways of thought and spiritual achievement. For Zen is unique in the whole range of human understanding, and Dr. Suzuki is accepted as its greatest exponent. 
Amazon
  back

Papers

Polanyi, Michael, "Life transcending physics and chemistry", Chemical and Engineering News, 45, 35, 21 August 1967, page 54-66. back

Links

Albert Einstein, The Problem of Space, Ether and Field in Physics, ' It is clear that the concept of space as a real thing already existed in the extra- scientific conceptual world. Euclid’s mathematics, however, knew nothing of this concept as such; they confined themselves to the concepts of the object, and the spatial relations between objects. The point, the plane, the straight line, length, are solid objects idealized. All spatial relations are reduced to those of contact (the intersection of straight lines and planes, points lying on straight lines etc). Space as a continuum does not figure in the conceptual system at all. This concept was first introduced by Descartes, when he described the point-in-space by it co-ordinates. Here for the first time geometrical figures appear, up to a point, as parts of infinite space, which is conceived as a three-dimensional continuum.' back

Aristotle, Physics V, iii, ' A thing that is in succession and touches is 'contiguous'. The 'continuous' is a subdivision of the contiguous: things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are two. This definition makes it plain that continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one, e.g. by a rivet or glue or contact or organic union. ' 227a10 sqq back

Bible, Bible: King James Version, 'About the Bible, King James Version The original electronic text for this version of the Bible was provided by the Oxford Text Archive. Original tagging was performed by the New Centre for the Oxford English Dictionary (Waterloo). Subsequent conversion to SGML was performed by the University of Michigan Humanities Text Initiative. The HTI is grateful for the permission of the Oxford Text Archive to provide access to the text.' back

Cantor's theorem - Wikipedia, Cantor's theorem - Wikipedia, the free encyclopedia, ' In elementary 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, denoted by P(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 members has a total of 2n subsets, so that if card (A) = n, then card (P(A)) = 2 n , and the theorem holds because 2n > n for all non-negative integers. ' back

Charles Darwin - Wikipedia, Charles Darwin - Wikipedia, the free encyclopedia, 'Charles Robert Darwin FRS (12 February 1809 – 19 April 1882) was an English naturalist. He established that all species of life have descended over time from common ancestry, and proposed the scientific theory that this branching pattern of evolution resulted from a process that he called natural selection.' back

Chemi Shalec, Netanyahu May Be Just as Corrupt and Disruptive, but He's No Match for Trump in Terms of Crazy, ' The Israeli prime minister’s spurts of destructive derangement, however, aren’t in the same league as Trump’s overarching and comprehensive dementedness. In terms of irrationality, Trump is the real McCoy. Which is why Netanyahu would never dream of ignoring the coronavirus, imploring his country to do the same, or intentionally and actively exposing his fans to the virus – for the time being, at least. For Netanyahu, it’s still a bridge too far; for Trump, just another way station on the route to the mad, mad, mad, mad world of his dreams and desires.' back

Claire Corbould, Towards Wakanda - Chadwick Boseman's passing and the limits of Afrofuturism, ' If you’re not a comics fan, you may have been surprised at the extent of the heartfelt grief expressed following the death of actor Chadwick Boseman. One explanation lies in the extraordinary power of the 2018 movie Black Panther, in which Boseman starred as T’Challa/Black Panther, to address racist stereotypes about Africa and Africans. Boseman’s character was heir to the hidden kingdom of Wakanda, a mythical African nation free of European colonisation. The film’s subtext explores African Americans’ varying identifications, past and present, with Africa and a global Black diaspora.' back

Code of Canon Law, 252, The formation of clerics, 'Can. 252 §1. Theological instruction is to be imparted in the light of faith and under the leadership of the magisterium in such a way that the students understand the entire Catholic doctrine grounded in divine revelation, gain nourishment for their own spiritual life, and are able properly to announce and safeguard it in the exercise of the ministry. §2. Students are to be instructed in sacred scripture with special diligence in such a way that they acquire a comprehensive view of the whole of sacred scripture. §3. There are to be classes in dogmatic theology, always grounded in the written word of God together with sacred tradition; through these, students are to learn to penetrate more intimately the mysteries of salvation, especially with St. Thomas as a teacher. There are also to be classes in moral and pastoral theology, canon law, liturgy, ecclesiastical history, and other auxiliary and special disciplines, according to the norm of the prescripts of the program of priestly formation.' back

Computability theory - Wikipedia, Computability theory - Wikipedia, the free encyclopedia, 'Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory. The basic questions addressed by recursion theory are "What does it mean for a function from the natural numbers to themselves to be computable?" and "How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched.' back

Dan Akroyd, The Blues Brothers, back

Eliza Berlage and James Whitmore, Global count shows tree numbers have haved since dawn of human civilization, 'Peter Kanowski, a forestry professor at Australian National University, said “all sorts of forests are valuable for all sorts of reasons, and we’re losing too much of them”. “The scale and rate of loss of biodiversity, carbon stock and other forest values, that has been at unprecedented rates for much of the past 50 years, is impacting profoundly and perhaps irreversibly on ecosystems, livelihoods, forest values and climate at scales from global to local,” he said.' back

Exodus 32, Exodus 32:27, '27 Then he said to them, “This is what the Lord, the God of Israel, says: ‘Each man strap a sword to his side. Go back and forth through the camp from one end to the other, each killing his brother and friend and neighbor.’” 28 The Levites did as Moses commanded, and that day about three thousand of the people died. 29 Then Moses said, “You have been set apart to the Lord today, for you were against your own sons and brothers, and he has blessed you this day.”' back

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

Geodesic - Wikipedia, Geodesic - Wikipedia, the free encyclopedia, 'In mathematics, particularly differential geometry, a geodesic . . . is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be (locally) the shortest path between points in the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it. The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. . . . Geodesics are of particular importance in general relativity, as they describe the motion of inertial test particles.' back

George Weigel, The Vatican should speak up on China's repression in Hong Kong, back

Gregory Cowles, Oliver Sacks dies at 82; Neurologist and Author Explored the Brain's Quirks, 'In 1989, interviewing him for “The MacNeil/Lehrer NewsHour,” Joanna Simon asked Dr. Sacks how he would like to be remembered in 100 years. “I would like it to be thought that I had listened carefully to what patients and others have told me,” he said, “that I’ve tried to imagine what it was like for them, and that I tried to convey this. “And, to use a biblical term,” he added, “bore witness.” ' back

Hilbert space - Wikipedia, Hilbert space - Wikipedia, the free encyclopedia, 'The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.' back

History of Mathematics - Wikipedia, History of Mathematics - Wikipedia, the free encyclopedia, 'The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.' back

Invisible hand - Wikipedia, Invisible hand - Wikipedia, the free encyclopedia, 'In economics, the invisible hand is a metaphor used by Adam Smith to describe unintended social benefits resulting from individual actions. The phrase is employed by Smith with respect to income distribution (1759) and production (1776). The exact phrase is used just three times in Smith's writings, but has come to capture his notion that individuals' efforts to pursue their own interest may frequently benefit society more than if their actions were directly intending to benefit society.' back

Jack Waterford, Border Force fiasco calls into question culture andleadership in 'paramilitary' group, 'The Border Force Fiasco is much more than last Friday. It can be seen through a thousand prisms. As a study of leadership, accountability and responsibility. As yet another example of the Abbott government's haplessness, hopelessness and current incapacity to win a trick. About whether it is actually smart to stir up public anxieties about aliens, strangers and terrorists. About whether the ABF contains the sort of people to whom we ought to give guns, and, implicitly, the right to use them. About whether it has the checks and balances necessary whenever people are vested with power over the lives of others. back

Jason Wilson, Twitter tyrants and social justice warriors beware! The libertarians are coming for you, 'It’s almost as if the values the cultural libertarians articulate aren’t intended to be applied universally at all. They just seem to want free speech for themselves and their mates.' back

Jennifer Taub, Trump among the kleptocrats, ' The United States is a money-laundering mecca. Our legal system, corporate lawyers, bankers, real estate agents, title companies and accountants are eager to turn dirty money into gold. Or yachts. Or sparkling new luxury condos in Manhattan and South Florida. Though the true owners of these clean assets largely hide from view, the fact that America welcomes big dirty money from abroad is no secret. The mystery, however, is why our leaders in Washington have not taken the simple steps to stop this.' back

John Maynard, Capturing the lived history of the Aborigines Protection Board while we still can, '2015 marks 100 years since amendments to the NSW Aborigines Protection Act gave the board far-reaching powers with consequences that are felt to this day. The 1915 amendments gave the board complete power to remove Aboriginal children from their families. They also enabled the acceleration of the revocation of Aboriginal reserves and the casting off of Aboriginal families from largely independent and successful farms around the state. For many Aboriginal people of those times, this was a board not of protection, but persecution. The board utterly controlled the lives and affairs of Aboriginal people in NSW from 1883 until 1969.' back

John von Neumann, The 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

Meditations on First Philosophy - Wikipedia, Meditations on First Philosophy - Wikipedia, the free encyclopedia, 'Meditations on First Philosophy (subtitled In which the existence of God and the immortality of the soul are demonstrated) is a philosophical treatise written by René Descartes first published in Latin in 1641. . . . The book is made up of six meditations, in which Descartes first discards all belief in things which are not absolutely certain, and then tries to establish what can be known for sure.' back

Michael Pembroke, Chnia's rise: why US advocacy for confrontaion leaves Asia cold, ' In this “plague” year, while the rest of the world reels from an unwanted pandemic, China’s trade figures for June 2020 exceeded figures from June 2019 – seven months before the pandemic. The International Monetary Fund expects that the economies of almost every country in the world, except China , will experience a contraction in 2020. Its prediction for the US is minus 8 per cent. The longer-term disparity is more stark. In 2019, the IMF projected that, by 2024, China’s contribution to global growth would exceed 28 per cent, while that of the US would fall from 13.8 to 9.2 per cent. The Covid-19-induced global recession is unlikely to change the trajectory.' back

New York Times Editorial, Australia's Brutal Treatment of Migrants, 'Prime Minister Tony Abbott has overseen a ruthlessly effective effort to stop boats packed with migrants, many of them refugees, from reaching Australia’s shores. His policies have been inhumane, of dubious legality and strikingly at odds with the country’s tradition of welcoming people fleeing persecution and war.' back

Noether's theorem - Wikipedia, Noether's theorem - Wikipedia, the free encyclopedia, 'Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.' back

Oliver Passon, What you always wanted to know about Bohmian mechanics but were afraid to ask, ' Bohmian mechanics is an alternative interpretation of quantum mechanics. We outline the main characteristics of its non-relativistic formulation. Most notably it does provide a simple solution to the infamous measurement problem of quantum mechanics. Presumably the most common objection against Bohmian mechanics is based on its non-locality and its apparent conflict with relativity and quantum field theory. However, several models for a quantum field theoretical generalization do exist. We give a non-technical account of some of these models.' 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 Hannam, Australia's carbon emissions extend rebound, led by coal fired power, 'Greenhouse gas emissions from Australia's main sources of energy use jumped by almost 10 million tonnes in the first year after the end of the carbon price, making it harder for the Abbott government to meet its goals of reducing carbon pollution. Total annualised carbon emissions in the 2014-15 financial year from the National Electricity Market (NEM) supplying about 80 per cent of the Australian consumers, all petroleum products consumed nationally and natural gas use in south-east Australia, rose by 9.6 million tonnes, or 3.3 per cent, consultancy Pitt & Sherry said. The contribution from the NEM alone rose by 6.4 million tonnes in the year as brown and black coal use in the power sector jumped, while transport-sourced emissions rose 1.8 million tonnes, and gas for use other than in electricity generation added 1.4 million tonnes, the firm's latest Cedex report found. back

Philip Oltermann, How Angela Merkel's great migant gamble paid off, ' By 2017, there was a prevalent view that Wir schaffen das would be Merkel’s undoing, a “catastrophic mistake” as Donald Trump said in January that year. “The worst decision a European leader has made in modern times,” Nigel Farage told Fox News. “She’s finished.” Yet today Merkel still sits at the top of Europe’s largest economy, her personal approval ratings back to where they were at the start of 2015 and the polling of her party, the Christian Democratic Union (CDU), buoyed to record levels by the global pandemic. When Merkel steps down ahead of federal elections in 2021, as is expected, her party’s successor currently looks more likely to be a centrist in her mould than a hardliner promising a symbolic break with her stance on immigration.' back

Problem of universals - Wikipedia, Problem of universals - Wikipedia, the free encyclopedia, 'The problem of universals is an ancient problem in metaphysics about whether universals exist. Universals are general or abstract qualities, characteristics, properties, kinds or relations, such as being male/female, solid/liquid/gas or a certain colour[1], that can be predicated of individuals or particulars or that individuals or particulars can be regarded as sharing or participating in. . . . The problem of universals is about their status; as to whether universals exist independently of the individuals of whom they can be predicated or if they are merely convenient ways of talking about and finding similarity among particular things that are radically different. This has led philosophers to raise questions like, if they exist, do they exist in the individuals or only in people's minds or in some separate metaphysical domain?' back

Quantum simulator - Wikipedia, Quantum simulator - Wikipedia, the free encyclopedia, 'Quantum simulators permit the study of quantum systems that are difficult to study in the laboratory and impossible to model with a supercomputer. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems. A universal quantum simulator is a quantum computer proposed by Richard Feynman in 1982. Feynman showed that a classical Turing machine would experience an exponential slowdown when simulating quantum phenomena, while his hypothetical universal quantum simulator would not. David Deutsch in 1985, took the ideas further and described a universal quantum computer. In 1996, Seth Lloyd showed that a standard quantum computer can be programmed to simulate any local quantum system efficiently' back

Quantum Turing machine - Wikipedia, Quantum Turing machine - Wikipedia, the free encyclopedia, 'A quantum Turing machine (QTM), also a universal quantum computer, is an abstract machine used to model the effect of a quantum computer. It provides a very simple model which captures all of the power of quantum computation. Any quantum algorithm can be expressed formally as a particular quantum Turing machine. Such Turing machines were first proposed in a 1985 paper written by Oxford University physicist David Deutsch suggesting quantum gates could function in a similar fashion to traditional digital computing binary logic gates' back

Richard Feynman, Lectures on Physics, volume 1 chapter 16, ' 16–4Relativistic mass We learned in the last chapter that the mass of an object increases with velocity, but no demonstration of this was given, in the sense that we made no arguments analogous to those about the way clocks have to behave. However, we can show that, as a consequence of relativity plus a few other reasonable assumptions, the mass must vary in this way.' back

Richard P. Feynman, Simulating Physics with Computers, 'I want to talk about the possibiity that there is to be an exact simulation, that the computer will do exactly the same as nature. If this is to be proved and the type of computer is as I've already explained, then it's going to be necessary that everything that happens in a finite volume of space and time would have to be exactly analyzable with a finite number of logical operations. The present theory of physics is not that way, apparently. It allows space to go down into infinitesimal distances, wavelengths to get infinitely great, terms to be summed in infinite order, and so forth; and therefore if this proposition is right, physical law is wrong.' back

Rick Noack, Iranian musician Mehdi Rajabian to stand trial for collaborating with female artistd, ' On Saturday, Rajabian is set to stand trial for working with female artists, according to activists who are monitoring his case. One of the performers in question — dancer Helia Bandeh, who performed to his music in a video published online — is based in the Netherlands, beyond the reach of Iran’s justice system. Even after serving previous prison terms over charges related to his work, Rajabian has approached the upcoming trial with defiance. “I strongly believe in the philosophy and message of music, artistic independence and an uncensored world,” Rajabian said in a text messages from the Iranian city of Sari, after posting bail. back

Sanya Dosanu & Chai Dingari, The Three Rules of Coronavirus Communication, ' Imagine living in a country where, six months into a pandemic, you aren’t totally confused. . . . For millions around the world, this isn’t a fantasy — it’s their reality. While Americans are still fighting over masks, New Zealand, Vietnam and Rwanda, to name a few, used clear and consistent public health messaging to build trust. This made it possible to stem their initial outbreaks and control new cases as they popped up. In the video above, we break down the simple yet highly effective rules of communication these countries deployed.' back

Sheffer stroke - Wikipedia, Sheffer stroke - Wikipedia, the free encyclopedia, 'In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" . . . denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or the alternative denial, since it says in effect that at least one of its operands is false.' back

Sidney Leng, China censors Thomas Piketty's book that touches on nation's growing inequality, ' But Piketty’s new book Capital and Ideology, which expands on the theme of inequality, looks increasingly unlikely to have the same success after falling foul of China’s censors. Published outside China last year, it has yet to be launched in the mainland due to demands from Piketty’s Chinese publisher, Citic Press Group, that all parts of the book related to inequality in China be cut. “I refused these conditions, so at this stage it looks as if Capital and Ideology won’t be published in China,” Piketty told the South China Morning Post by email.' back

Slezak, Doman, Shatoba, Timms & Palmer , The Mystery of the Murray Darling's vanishing flows, ' More than 2 trillion litres of water — enough to fill Sydney Harbour four and a half times — has gone missing from our largest and most precious river system — the Murray-Darling Basin. . . . According to an investigation by some of Australia’s top water scientists, shared exclusively with the ABC, 20 per cent of the water expected to flow down the rivers from 2012-2019 was simply not there. That’s despite almost $7 billion being spent to protect the health of the system’s rivers and ecosystems that rely on them.' back

Spencer Ackerman, West Point professor calls on US military to target legal critics of war on terror, 'An assistant professor in the law department of the US Military Academy at West Point has argued that legal scholars critical of the war on terrorism represent a “treasonous” fifth column that should be attacked as enemy combatants. In a lengthy academic paper, the professor, William C Bradford, proposes to threaten “Islamic holy sites” as part of a war against undifferentiated Islamic radicalism. That war ought to be prosecuted vigorously, he wrote, “even if it means great destruction, innumerable enemy casualties, and civilian collateral damage”.' back

Tess (1979 film) - Wikipedia, Tess (1979 film) - Wikipedia, the free encyclopedia, ' Tess is a 1979 drama film directed by Roman Polanski and starring Nastassja Kinski, Peter Firth, and Leigh Lawson. It is an adaptation of Thomas Hardy's 1891 novel Tess of the d'Urbervilles. The screenplay was written by Gérard Brach, John Brownjohn, and Roman Polanski. The film received positive critical reviews upon release and was nominated for six Academy Awards, including Best Picture, winning three for Best Cinematography, Best Art Direction and Best Costume Design.' back

Thomas L Friedman, Our Radical Islamic BFF, Saudi ArBIA, ' "“The last few decades have seen this attempt to homogenize Islam,” claiming “there is only one legitimate path to God,” Haqqani said. And when there is only one legitimate path, “all others are open to being killed. That has been the single most dangerous idea that has emerged in the Muslim world, and it came out of Saudi Arabia and has been embraced by others, including the government in Pakistan.” ' back

Thomas Robert Malthus - Wikipedia, Thomas Robert Malthus - Wikipedia, the free encyclopedia, 'The Reverend Thomas Robert Malthus FRS (February 1766 – December 1834) was an English scholar, influential in political economy and demography. Malthus popularized the economic theory of rent. Malthus has become widely known for his theories about population and its increase or decrease in response to various factors. The six editions of his An Essay on the Principle of Population, published from 1798 to 1826, observed that sooner or later population gets checked by famine and disease. ' back

Tom Heenan, Seventy years after Hiroshima, who was Australian wae correspondent Wilfred Burchertt?, 'Burchett wrote stories that the Australian and US governments preferred not to be told and paid the price. He covered wars in which Australians fought on the other side. He was not “a my country right or wrong” barracker, but reported the facts as he saw them, and for the most part got them right. His career should be judged on all his achievements and not reduced to a solitary story.' back

Walid Aly, Dyson Heydon: the pub test is a tool in the politics of cynicism, 'This isn't a mere question of semantics. Let's be clear about this: when Bandt or Oliver appeal to pub sensibilities to suggest it is some obvious howler that Heydon would hear an application against himself, they are in fact attacking the normal operation of the legal system. In making his judgment, Heydon was acting as countless judges have for countless years. As you read this right now, there's a fair chance a judge somewhere in Australia is doing the same in a very important case involving lots of money or the fate of some children after a divorce.' back

Woomera (spear-thrower) - Wikipedia, Woomera (spear-thrower) - Wikipedia, the free encyclopedia, ' A woomera is a wooden Australian Aboriginal spear-throwing device. Similar to an atlatl, it serves as an extension of the human arm, enabling a spear to travel at a greater speed and force than possible with only the arm. ' back

www.naturaltheology.net is maintained by The Theology Company Proprietary Limited ACN 097 887 075 ABN 74 097 887 075 Copyright 2000-2020 © Jeffrey Nicholls