Natural Theology

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Essay 34: Empire is lethal and unnatural and it will eventually die

Here and elsewhere we shall not obtain the best insights into things until we actually see them grow from the beginning (Aristotle quoted in Jaeger) Werner Jaeger (1997): Aristotle: Fundamentals of the history of his development.

Things are getting desperate so we have no time to muck around, we must get straight to the point. The concept of empire contradicts quantum mechanics; quantum mechanics is the foundation of the universe; therefore empire contradicts the universe.

Empires built on military violence have reshaped the human world over the last ten thousand years at enormous costs in life, language and culture.

In a recent article Massimo Faggioli explains that American Catholicism is no longer just a refuge for socially conservative ideologues as it was between the 1990s and the early 2000s. It is now a brand, for sale to the highest bidder. Massimo Faggioli (2025_02_19): Donald Trump captured American Catholicism — and the ramifications are being felt around the world

Like half the population of the US, the Church has sold out to the king in waiting while the revolutionary Pope Francis lies dying. Francis brought the liberation theology of Latin America to Rome but he met stiff opposition from within. This was to be expected, given the history of the Church.

We are in a battle between empire and nature. On the one hand we have imperialism, represented by Pope Francis and would be king Donald. Although Francis is dying, but he represents a very well defined and ancient imperial force. It s unlikely that his successor will be much different.

Trump is alive and kicking. He does not appear to know what he is doing but his masters are imperialists and they have just one idea, to become wealthy and powerful at the expense of everybody else.

The imperial Roman Catholic Church

We begin with the papal backstory.

The idea of an omnipotent and omniscient controller goes a long way back. We see it recorded in literary classics like the Mahabharata. It is probably related to the invention of writing. In the old days we can imagine two classes of writers: accountants represented by thousands of clay tablets inscribed with cuneiform inventories; and scribes and poets working for powerful politicians to create legends for their masters to explain and justify their power and to convey their commands to their subjects. Zahiru'd-din Mohammad Babur (2020): The Babur Nama

Most everyday people who grew food, built roads, made jewellery, art, weapons, harnesses, vehicles, cities, pyramids, mines and aqueducts were illiterate so they did not write. Their stories did not surface until much later. Subtle stories about nature only really came to the surface after the scientific method, based on physical contact with reality rather than trade and politics, became established.

Jesus of Nazareth was a good man, a Jew who knew the Law and quoted it.

A Pharisee asked him which commandment in the law is the greatest?”

Jesus said to him, “You shall love the Lord, your God, with all your heart, with all your soul, and with all your mind. This is the greatest and the first commandment.

The second is like it: You shall love your neighbour as yourself. The whole law and the prophets depend on these two commandments.” (Mt 22: 36-40)

He was also a radical who attacked the religious establishment in Judea: You brood of vipers! How can you speak good, when you are evil? For out of the abundance of the heart the mouth speaks. (Mt 12:34).

This upset the ruling priests and they colluded with the occupying Romans to have him crucified. Jesus was brought before the Governor, Pontius Pilate, who questioned him and declared him innocent.

But the populists in the crowd shouted, “Crucify him! Crucify him! (Lk 23:21). So Pilate made a very Trumplike move: he released the man who had been imprisoned for rebellion and murder, the popular choice, and he handed Jesus over to the crowd for execution. Jesus was crucified in the Roman manner, but legend has it that he did not stay dead but came alive again on the third day.

Once again the scribes created a new world. The Hebrew Bible tells a story similar to many ancient texts, a difficult relationship between a god and their people, but promise of a glorious future for the nation of Israel. This did not happen. Superior imperial forces crushed them. The Roman occupation of Judea ended it with the expulsion of the Israel from their promised land.

The writers of the Christian New Testament used the life of Jesus to devise a happy ending to this sadness. They revisited the catastrophic human interaction with Yahweh and Satan in the Garden of Eden. They explained that the death of Jesus, the Son of Yahweh has placated the old man for the original insult. Death was conquered by his resurrection, baptism qualified his followers to an eternal life of post mortem bliss.

By the fourth century these appealing features of Christianity made it a force to be reckoned with.

The emperor Constantine, motivated by his mother (now Saint) Helena, saw an opportunity exploited a millennium later by the English King Henry VIII. He established Christianity as the imperial religion and expropriated pagan property.

He convened a council of Christian bishops in Nicaea (now İznik, Turkey) which met from May until the end of July 325. The Council asserted that Jesus really was the Son of God and drafted the Nicene Creed as a symbol of Christian unity. The creed makes no mention of the law of love promulgated by Jesus.

In 381 the Council of Constantinople further standardized Christian doctrine, and confirmed and expanded the Nicean creed into the Niceno-Constantinopolitan creed still recited today. The Fathers of the Church produced a vast literature refining the details of Christianity. Aurelius Augustinus, the bishop of Hippo in Roman North Africa, explained the Trinity established by the Councils. His ideas have since been taken up by many theologians.

All this converted Christianity into the Roman Catholic Church. The Church grew in power and popularity. It survived the fall of the Empire to become an autocratic theological military organization. It executed a long series of crusades against Islam and other “heresies”. It remained the centre of political power in Europe until the French Revolution

This Church enjoyed a brief scientific episode in the middle Ages, partly as a consequence of the Crusades. The work of Aristotle had been preserved in the Eastern Empire. It came into Islamic hands during the Muslim-Byzantine wars. Aristotle’s Greek was translated into Arabic and many Muslim scholars wrote commentaries on his work. Thomas Aquinas and others adapted Aristotle’s work to service Christianity. They developed a model of God based on the unmoved mover described in book XII of Aristotle’s Metaphysics. Michael Bordt (2011): Why Aristotle's God is not the Unmoved Mover

The military operations of the Church were very expensive. To pay for them the Church began to sell indulgences. The faithful could make cash payments for posthumous reductions of the punishment in purgatory required for entry to heaven.

This abuse was noticed by Martin Luther among others and led to a major split in the Church, the Reformation.

In semiliterate Europe the Bible was effectively a secret document, written in Greek, Hebrew and Aramaic.

Luther translated it into German. Printing was invented and millions of bibles began to circulate. People to see for themselves the scriptural foundations of the Church. This unsettled the church which had previously maintained a monopoly on interpretation of the Bible.

Then the Church lost a scientific battle with Galileo about the structure of the solar system, the Galileo affair. Galileo was right.

The Holy Inquisition gave him the option of recantation to avoid the death penalty for heresy, and he wisely chose it. He had seen the truth with his telescope and new that staying alive would cause no harm. Galileo Galilei (1633): Recantation of Galileo (June 22, 1633)

On the centenary of Einstein’s birth on 10 November 1979 the Roman Pope Saint John Paul II foreshadowed a modern study of this “Galileo Affair”. On 3 July 1981 the Papal Academy of Science established a Study Commission for this purpose. The Pope announced its conclusions in an address to a Plenary Session of the Academy on 31 October 1992:

4. [. . .] One might perhaps be surprised that, at the end of the Academy’s study week on the subject of the emergence of complexity in the various sciences, I am returning to the Galileo case. Has not this case long been shelved and have not the errors committed been recognised?

That is certainly true. However, the underlying problems of this case concern both the nature of science and the message of faith. It is therefore not to be excluded that one day we shall find ourselves in a similar situation, one which will require both sides to have an informed awareness of the field and of the limits of their own competencies. Saint John Paul II (1992_10_31): Address to the Plenary Session on ‘The Emergence of Complexity in Mathematics, Physics, Chemistry and Biology’

Throughout his discussion of the Church’s treatment of Galileo the Pope stuck firmly to the ecclesiastical dichotomy between revelation and science.

He explains:

12. [. . .] There exist two realms of knowledge, one which has its source in Revelation and one which reason can discover by its own power. To the latter belong especially the experimental sciences and philosophy. The distinction between the two realms of knowledge ought not to be understood as opposition. The two realms are not altogether foreign to each other; they have points of contact. The methodologies proper to each make it possible to bring out different aspects of reality [. . .].

14.  Humanity has before it two modes of development. The first involves culture, scientific research and technology, that is to say whatever falls within the horizontal aspect of man and creation, which is growing at an impressive rate. [. . .] The second mode of development involves what is deepest in the human being, when, transcending the world and transcending himself, man turns to the One who is the Creator of all. It is only this vertical direction which can give full meaning to man’s being and action, because it situates him in relation to his origin and his end.

And so we reach the 21st century Roman Catholic Church that sold itself to the kinglet Donald. The Church is an absolute monarchy ruled by an infallible Pope. It derives its legitimacy from an invisible imaginary God whose properties are mutually contradictory.

In it women are second class citizens without the spiritual qualities necessary to join the Priesthood. Their bodies belong to God to create more Catholics. The children are indoctrinated from a tender age. Meanwhile clergy around the world sexually abuse children and the Church vigorously resists pressure to do anything about it.

In order to keep itself pure it promotes a doctrine of two truths, science and philosophy on one hand, divine revelation on the other, curated by its own tame theologians.

It remains true that most people will believe anything in order to avoid execution, or even to avoid losing their jobs. In order to free ourselves from the need to believe absurdities, we must replace this ancient theology with something scientific based on reality.

What has Nature got to offer?

What has nature got to offer against this magnificent and ancient structure of art, architecture, literature and a committed following of more than a billion people? The short answer is, everything. We begin with the hypothesis that the world itself is divine. From this point of view everything from the creation of the world to the most sublime reults of human art and science are natural products.

The essence of scientific method is physical contact with the world. As John Hopfield noted in the lecture he gave when he won the Nobel Prize for his role in the development of artificial intelligence, physics is a point of view for understanding the totality of humanity and our Universe. John J. Hopfield (2024_12_08): Nobel Lecture: Physics is a point of view

The history of many fields is marked by episodes of rapid change between longer periods of business as usual. Thomas Kuhn noticed this pattern and his work attracted a lot of interest from historians of science. Thomas Kuhn (1996): The Structure of Scientific Revolutions

The most salient feature of any revolution is that some of the old system must go. In the political sphere, what has to go is imperialism, the claim of a certain subset of the human population to command, exploit, enslave and kill other people.

Jesus put it clearly: love God, now the whole universe; love your neighbour, that is everybody, including Samaritans.

These ideas have emotional appeal but they have deeper scientific significance. Because the universe plays all the roles of divinity, physics and theology have the same subject and must be mutually compatible. Science feeds on physical contact. Theology must see and feel god if it is to be a science. Gravity is our primary contact with divinity and shapes our world.

Popes think they can be infallible because they have an error free connection to a God who, like Laplace’s demon, controls every event at every moment everywhere in the world. This is impossible. The airline industry is populated with people who go over the top trying to be infallible. Despite their efforts, every few billion kilometres something goes wrong. If it is bad enough, it makes the news but there are lot of near misses to keep them on their toes.

We began with a little syllogism: The concept of empire contradicts quantum mechanics; quantum mechanics is the foundation of the universe; therefore empire contradicts the universe.

Now we must see how quantum mechanics builds universe. From this it becomes clear that imperialism can only be maintained by predatory military violence.

Quantum mechanics is itself the result of global paradigm change, the loss of determinism.

For me one of the saddest stories in science is the career of Albert Einstein. He started brilliantly in 1905 and revolutionized physics: special relativity; E = mc2; the light quantum, for which he won a Nobel prize nearly twenty years later. In 1915 he produced the general theory of relativity. This work established his reputation forever and showed us the universe as a whole.

But he became doubtful about quantum mechanics. He produced some trenchant critiques which forced the proponents of the new theory to think deeply. One of these, the EPR paper, brought attention to spooky action at a distance, entanglement. This phenomenon shows that quantum mechanics lies deep in the foundation of the world, before the advent of space and time. Einstein, Podolsky & Rosen (1935): Can the Quantum Mechanical Description of Physical Reality be Considered Complete?

In 1933 he gave the Herbert Spencer Lecture at the University of Oxford. This lecture marks a definitive break with quantum theory. Speaking of the work of de Broglie, Schrödinger, Dirac and Born he said:

I cannot help confessing that I myself accord to this interpretation no more than a transitory significance. I still believe in the possibility of giving a model of reality, a theory, that is to say, which shall represent events themselves and not merely the probability of their occurrence. Albert Einstein (1933): On the Method of Theoretical Physics: Herbert Spencer Lecture 1933

This was not to be. Quantum mechanics does in fact define mathematical equations which model of reality, but these equations are multi-dimensional and describe a probabilistically distributed spectrum of results for any experiment. These probabilities upset him:

Die Quantenmechanik ist sehr achtung-gebietend. Aber eine innere Stimme sagt mir, daß das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, daß der nicht würfelt.

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one." I, at any rate, am convinced that He does not throw dice. Albert Einstein (1926): God does not throw dice

He was a determinist, a follower of the demon Laplace desribed in his Philosophical Essay on Probabilities:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. Laplace's demon - Wikipedia

This is an ancient belief. The omniscient and omnipotent God of Christianity, for instance has immediate providence over everything. Every event is known to God and ‘meant to be’. Aquinas, Summa, I, 22, 3: Does God have immediate providence over everything?

Charles Darwin was not the first to think of evolution, but he recognized that the essential ingredient of evolutionary creativity is variation. He studied it in detail. Living in a rural community, he saw how the deliberate selection of breeding stock enabled farmers to gradually move their livestock in particular directions, toward faster horses, sweeter apples or finer wool. Charles Darwin (1875): The Variation of Animals and Plants Under Domestication

A deterministic process cannot be creative. A reliable computer presented with a particular problem gives the same answer every time. This concept is embodied in the software engineering mantra garbage in garbage out . This has become newly relevant with the artificially intelligent machines that trawl the internet for guidance. Judith Bishop (2025_02_26): Erotica, gore and racism: how America’s war on ‘ideological bias’ is letting AI off the leash

Without the probabilistic feature of quantum mechanics, it does not seem possible for the Universe to have evolved from a structureless, eternal, omnipotent initial singularity to its present state.

The conspicuous casualty of quantum mechanics which upset Einstein was the loss of determinism, but this was only partial. Although the timing of particular quantum events is unpredictable the nature of the events is defined with great precision. The quantum of action, the masses of particles, the speed of light and the values of quantum states measured by energy or frequency are precisely related by the equations E = ℏ𝜔 and m = E / c 2. W. F. McGrew et al: Atomic clock performance enabling geodesy below the centimetre level

The early days of quantum theory from 1900 to the early 1920s were marked by incremental changes associated with the names of Planck, Einstein, Bohr, de Broglie, Heisenberg and Schrödinger. After Einstein’s revelation of Minkowski space it was obvious that quantum theory must conform to special relativity.

The first important step in this direction was taken by Paul Dirac. He began with the non-relativistic Schrödinger equation in space time which is linear in time but quadratic in space / momentum. Special relativity treats space and time on an equal footing and Dirac felt that time must be linear, so he needed to find the square root of the momentum operator. He found a neat trick with the gamma matrices to eliminate the cross terms in this square root and arrived at the Dirac equation. He wrote:

It was found that the equation gave the particle a spin of half a quantum. And also gave it a magnetic moment. It gave just the properties that one needed for an electron. That was really an unexpected bonus for me, completely unexpected.

Dirac’s equation also predicted the existence of antimatter, although this was not clarified until Carl Anderson discovered positrons in cosmic rays. Carl D. Anderson (1936_12_12): Nobel Lecture: The production and properties of positrons

Dirac’s equation became a foundation of the relativistic quantum mechanics we call quantum field theory (QFT). This theory struck many difficulties. The principle problems arise from the assumption that spacetime is a continuous manifold. This problem was solved with renormalization and by the 1970s QFT had entered the mainstream of physical theory as a comprehensive description of the fundamental particles known as the Standard Model. Standard model - Wikipedia

Although many physicists are very proud of this achievement and some feel that they are approaching the mind of God, some philosophers and other critics are not so sure.

After an extensive analysis of QFT the philosopher Meinard Kuhlmann wrote:

In conclusion one has to recall that one reason why the ontological interpretation of QFT is so difficult is the fact that it is exceptionally unclear which parts of the formalism should be taken to represent anything physical in the first place. And it looks as if that problem will persist for quite some time. Meinard Kuhlmann (Stanford Encyclopedia of Philosophy): Quantum Feld Theory

Although QFT is an effective algorithm for working with the electromagnetic, weak and strong forces, gravitation is unrenormalizable so QFT fails to embrace the most ubiquitous force of all. From this point of view the standard model of physics is not much help in devising a suitable model for a divine universe.

Despite this difficulty, we now have classical interferometers to detect and measure the gravitational waves caused by cosmic events, a new window into the dynamics of the universe.

I noted above that intellectual revolutions often require old beliefs to be abandoned. Since is birth 125 years ago, one of the principal difficulties with the interpretation of quantum theory has been to break free of Minkowski space.

In their book on axiomatic QFT, Streater and Wightman write:

Since in quantum mechanics observables are represented by hermitian operators which act on the Hilbert space of state vectors, one expects the analogue in relativistic quantum mechanics of a classical observable field to be a set of hermitian operators defined for each point of space-time and having a well-defined transformation law under the appropriate group. Streater & Wightman (2000):PCT, Spin, Statistics and All That

In other words, we are building QFT on top of Minkowski space which as Einstein points out in at the end of his 1915 paper, is also assumed to be the foundation of gravitation:

However, the postulate of general relativity cannot reveal to us anything new and different about the essence of the various processes in nature than what the special theory of relativity taught us already. Albert Einstein (1915): The Field Equations of Gravitation

We notice that Minkowski space is pixellated by the quantum of action: ∆x . ∆p ≈ ∆E . ∆t ≈ ℏ. This suggests that the nature of discrete elementary particles has something to do with the structure of space.

Perhaps the physicists have got cart before the horse. Maybe Minkowski space is not the foundation of quantum mechanics, but a consequence.

I approach this problem from a theological direction, seeking to explore the hypothesis that the universe creates itself, an idea implicit in the hypothesis of an eternal omnipotent initial singularity. I am far from understanding the computational details of QFT but I am heartened by the words of Richard Feynman: physical understanding is completely unmathematical, imprecise, an inexact thing but absolutely necessary to a physicist; and Peter Osper: Research is to see what everybody has seen and think what nobody has thought. Feynman, Leighton and Sands FLP II_02: Chapter 2: Differential Calculus of Vector Fields, Peter Osper (1957): Review: Albert Szent-Györgyi (1957): Bioenergetics

In his Metaphysics written more than 2000 years ago Aristotle conceived an unmoved mover, an intellectual entity whose action is thought (Aristotle’s word energeia) to activate the formal universe of his mentor Plato 24 . The medieval Christian theologian Thomas Aquinas and his contemporaries transformed Aristotle’s work into a model of the Christian God.

In the Aquinas model, the creator is eternal, omnipotent, omniscient and absolutely simple. The attributes of omniscience and absolute simplicity reflect the ancient belief that intelligence is an immaterial power, so that the maximally intelligent being must also be maximally immaterial. In the modern world however we see, as Landauer tells us, that information is physical, represented by physical symbols like these letters. Rolf Landauer (1991): The physical nature of information

Let us start with an initial singularity, naked gravitation, analogous to the creator modelled by Aquinas. This entity is the substance of gravitation deprived of energy and the Minkowski space upon which Einstein built his field theory.

It is also deprived of the formal structure of omniscience postulated by Aquinas (since it has no physical structure to represent information). It also reflects the hypothesis that the total energy of the universe is zero. It is eternal (because nothing comes from nothing), and omnipotent since it is the source of the enormous universe. Its action is constrained, like the traditional Christian God, by the fact that inconsistent states cannot coexist. Richard Feynman (2002): Feynman Lectures on Gravitation

Further it fulfils the hypotheses of fixed point theory and acts at random since it has no structural control 28 29 . These conditions enable it to create a random Hilbert space within itself, that is variation, the creative feature of evolution. The mathematical idea of fixed point theory is similar to Augustine’s idea that the image formed when god reflects upon themself is in fact divine. Brouwer fixed point theorem - Wikipedia ,John Casti (1996): Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, Mary Sirridge (1999): Quam videndo intus dicimus: Seeing and Saying in De Trinitate XV

Quantum mechanics arises spontaneously as this random structure maps itself onto itself by superposition. We can imagine the emergence of stationary features in this wave field created by hermitian operators which correspond to the elementary particles we observe in the current universe.

We can further imagine that these stationary structures induce the bifurcation of naked gravitation into potential and kinetic energy. As in QFT the kinetic energy causes the instances of stationary quantum formalism to be realized as observable particles. The potential serves to bind the systems created together.

We observe that all the elementary particles fall into one of two classes, bosons and fermions. The properties of these particles may be seen as the sources of the Minkowski metric and their distinction accounts for its pixellation. The peculiar metric signature of Minkowski space, 1, 1, 1, -1, enables it to accommodate massless bosons travelling at the speed of light on null geodesics, and it also provides a three dimensional euclidean space for massive fermions, which are subject to the Pauli exclusion principle, to move freely.

This simple model frees quantum mechanics from the constraints of special relativity and provides a role for gravitation in the creation of the Universe. We know, from the modern conception of quantum mechanics as a theory of computation and communication, that it is well within its power, given a suitable evolutionary process, to create all the physical features of the Universe. Nielsen & Chuang (2016): Quantum Computation and Quantum Information

This simple model has features which suggest that democracy rules, at least in the world of elementary particles.

1. It provides a general framework which does not appear to contradict any of the data. It is not mathematical but may provide some of the insight recommended by Feynman:

A physical understanding is completely unmathematical, imprecise, an inexact thing but absolutely necessary to a physicist.

The idea is supported by Dirac’s feeling that mathematical success does not guarantee correctness:

The rules of renormalization give surprisingly, excessively good agreement with experiments. Most physicists say that these working rules are, therefore, correct. I feel that that is not an adequate reason. Just because the results happen to be in agreement with observation does not prove one's theory is correct. . . . I have spent many years searching for a Hamiltonian to bring into the theory and have not yet found it. I shall continue to work on it as long as he can, and other people I hope will follow along such lines Peter Goddard (1998): Paul Dirac, The Man and His Work

2. The creation of particles through quantum mechanics coupled with the provision of energy from gravitation endows each particle with a hermitian operator (a quantum mind) which governs its interactions with other particles in a way analogous to the role of our mental states in our interactions with one another.

3. Provision of 2 above eliminates the need for field, since particles carry the software to resolve all their one-to-one interactions. The infinity, continuity and lack of a clear ontological basis for fields creates most of the problems listed by Kuhlmann. Even before Gödel and Turing destroyed his formalist dream, Hilbert clearly rejected the reality of infinity which is implicit in field theory:

In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought — a remarkable harmony between being and thought. David Hilbert (1925): On the Infinite

4. A physical field is in effect a surrogate for imperial rule, subjecting the behaviour of all physical particles to some mysterious controller inserted from outside. Quantum field theory no more provides us with an origin for the fields that rule the world than the average imperialist can provide justification for any claim (other than violence) that they are entitled to control the behaviour of their “subjects”. The philosopher Sunny Auyang describes the rather confusing role of fields in QFT. Our position contrasts with the standard view that the fundamental reality of the world is field and attributes autonomy to particles. Sunny Auyang (1995): How is Quantum Field Theory Possible?

5. QFT cannot deal with gravitation because it cannot be renormalized and renormalizaztion is absolutely essential to QFT.

I conclude by emphasizing two principles that lie at the heart of my exposition and use them to construct a simple political model of the world based on quantum theory that holds at all scales.

The first I call the heuristic of simplicity. Here we are working at the base of the Universe right next to the initial singularity. Although theologians talk about the ineffable mysteries of God, if God is really absolutely simple there is nothing to be said about them except that they exist. As Aquinas puts it, the essence of God is their existence. The true story must begin quite simply.

As our picture of the universe becomes more complex, we are still subject to the information theoretical constraint that the entropy of an explanation should be equivalent to the entropy of the system described. This effects the courtroom injunction to tell the truth, the whole truth and nothjng but the truth. We expect the complexity of explanation to be equivalent to the complexity of the reality explained.

The second is the symmetry with respect to complexity built into quantum theory by unitarity. Communication theory shows that the sum of the probabilities of the phonemes in human speech or the symbols in human writing is 1, identical to the sum of the probabilities of the outcomes of a quantum observation. This probabilistic symmetry applies to all forms of communication, regardless of its complexity

This idea is implicit in quantum theory through the normalization of vectors independently of the dimension of the operative Hilbert space so the vector representing the universe comprising the superposition countable infinity of basis states representing (say) fundamental particles has length one and direction defined by the Universe.

Although the difference in scale between fundamental particles and an ideal democratic polity is immense, they are formally quite similar.

This similarity enables us to draw an analogy between the autonomous freedom of individual people and the autonomous freedom of individual elementary particles. Democratic politics, like quantum mechanics, fits in Hilbert space. Voting itself is linear, a form of superposition. Individuals and political parties are characterized by their directions in political space which may be modelled as vectors in a Hilbert space. Every person is represented by a basis vector and their sums in various combinations present us with a comprehensive picture of the political directions in an electorate. There seems to be fundamental harmony between democratic politics and the nature of the world.

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Further reading

Books

Auyang (1995), Sunny Y., How is Quantum Field Theory Possible?, Oxford University Press 1995 Jacket: 'Quantum field theory (QFT) combines quantum mechanics with Einstein's special theory of relativity and underlies elementary particle physics. This book presents a philosophical analysis of QFT. It is the first treatise in which the philosophies of space-time, quantum phenomena and particle interactions are encompassed in a unified framework.' 
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Babur (2020), Zahiru'd-din Mohammad, and Annette Susannah Beveridge (Translator) and William Dalrymple (Introduction), The Babur Nama, Knopf / Penguin Random House 2020 Jacket: 'Zahiru'd-din Mohammad Babu (1485-1530) a poet-prince from central Asia, was the first Mughal emperor and author of one of the most remarkable autobiographies in world Literature. The Babur Nama reveals its author as not only a military genius but also a ruler unusually magnanimous for his time, cultured, witty and possessing a talent for poetry, an adventurous spirit and an acute eye for natural beauty.' 
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Casti (1996), 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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Darwin (1875), Charles, and Harriet Ritvo (Introduction), The Variation of Animals and Plants Under Domestication (Foundations of Natural History), Johns Hopkins University Press 1875, 1998 ' "The Variation, with its thousands of hard-won observations of the facts of variation in domesticated species, is a frustrating, but worthwhile read, for it reveals the Darwin we rarely see -- the embattled Darwin, struggling to keep his project on the road. Sometimes he seems on the verge of being overwhelmed by the problems he is dealing with, but then a curious fact of natural history will engage him (the webbing between water gun-dogs' toes, the absurdly short beak of the pouter pigeon) and his determination to make sense of it rekindles. As he disarmingly declares, 'the whole subject of inheritance is wonderful.'. 
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Feynman (2002), Richard, Feynman Lectures on Gravitation, Westview Press 2002 ' The Feynman Lectures on Gravitation are based on notes prepared during a course on gravitational physics that Richard Feynman taught at Caltech during the 1962-63 academic year. For several years prior to these lectures, Feynman thought long and hard about the fundamental problems in gravitational physics, yet he published very little. These lectures represent a useful record of his viewpoints and some of his insights into gravity and its application to cosmology, superstars, wormholes, and gravitational waves at that particular time. The lectures also contain a number of fascinating digressions and asides on the foundations of physics and other issues.' [zero-energy universe, pp 9 - 10]  
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Goddard (1998), Peter , and Stephen Hawking, Abraham Pais, Maurice Jacob, David Olive, and Michael Atiyah, Paul Dirac, The Man and His Work, Cambridge University Press 1998 Jacket: Paul Adrien Maurice Dirac was one of the founders of quantum theory and the aithor of many of its most important subsequent developments. He is numbered alongside Newton, Maxwell, Einstein and Rutherford as one of the greatest physicists of all time. This volume contains four lectures celebrating Dirac's life and work and the text of an address given by Stephen Hawking, which were given on 13 November 1995 on the occasion of the dedication of a plaque to him in Westminster Abbey. In the first lecture, Abraham Pais describes from personal knowledge Dirac's character and his approach to his work. In the second lecture, Maurice Jacob explains not only how and why Dirac was led to introduce the concept of antimatter, but also its central role in modern particle physics and cosmology. In the third lecture, David Olive gives an account of Dirac's work on magnetic monopoles and shows how it has had a profound influence in the development of fundamental physics down to the present day. In the fourth lecture, Sir Michael Atiyah explains the widespread significance of the Dirac equation in mathematics, its roots in algebra and its implications for geometry and topology.' 
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Jaeger (1997), Werner Wilhelm, Aristotle: Fundamentals of the history of his development, Oxford University Press 1997 Jacket: '"Aristotle was the first thinker to set up along with his philosophy a conception of his own position in history; he thereby created a new kind of philosophical consciousness, more responsible and inwardly complex. He was the inventor of the notion of intellectual development in time . . . ." In this classic study, Professor Jaeger profoundly altered the general view of Aristotle among philosophers and classical scholars. He showed that Aristotle was not uncompromisingly opposed to Plato, that he developed gradually, applying step by step his particular genius to the problems of his age.' 
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Kuhn (1996), Thomas S, The Structure of Scientific Revolutions, U of Chicago Press 1962, 1970, 1996 Introduction: 'a new theory, however special its range of application, is seldom just an increment to what is already known. Its assimilation requires the reconstruction of prior theory and the re-evaluation of prior fact, an intrinsically revolutionary process that is seldom completed by a single man, and never overnight.' [p 7]  
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Nielsen (2016), Michael A., and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2016 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002. 
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Streater (2000), 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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Albert Einstein (1915), The Field Equations of Gravitation, ' In two recently published papers I have shown how to obtain field equations of gravitation that comply with the postulate of general relativity, i.e., which in their general formulation are covariant under arbitrary substitutions of space-time variables. [. . .] With this, we have finally completed the general theory of relativity as a logical structure. The postulate of relativity in its most general formulation (which makes space-time coordinates into physically meaningless parameters) leads with compelling necessity to a very specific theory of gravitation that also explains the movement of the perihelion of Mercury. However, the postulate of general relativity cannot reveal to us anything new and different about the essence of the various processes in nature than what the special theory of relativity taught us already. The opinions I recently voiced here in this regard have been in error. Every physical theory that complies with the special theory of relativity can, by means of the absolute differential calculus, be integrated into the system of general relativity theory — without the latter providing any criteria about the admissibility of such physical theory' back

Albert Einstein (1926), God does not throw dice, ' Die Quantenmechanik ist sehr achtung-gebietend. Aber eine innere Stimme sagt mir, daß das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, daß der nicht würfelt.
Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one." I, at any rate, am convinced that He does not throw dice.' back

Albert Einstein (1933), On the Method of Theoretical Physics: Herbert Spencer Lecture 1933, ' It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience. back

Aquinas, Summa, I, 22, 3, Does God have immediate providence over everything?, ' I answer that, Two things belong to providence—namely, the type of the order of things foreordained towards an end; and the execution of this order, which is called government. As regards the first of these, God has immediate providence over everything, because He has in His intellect the types of everything, even the smallest; and whatsoever causes He assigns to certain effects, He gives them the power to produce those effects. Whence it must be that He has beforehand the type of those effects in His mind. As to the second, there are certain intermediaries of God's providence; for He governs things inferior by superior, not on account of any defect in His power, but by reason of the abundance of His goodness; so that the dignity of causality is imparted even to creatures.' back

Brouwer fixed point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Among hundreds of fixed-point theorems] Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology.' back

Carl D. Anderson (1936_12_12), Nobel Lecture: The production and properties of positrons, ' To differentiate with certainty between the particles of positive and negative charge it was necessary only to determine without ambiguity their direction of motion. To accomplish this purpose a plate of lead was inserted across a horizontal diameter of the chamber. The direction of motion of the particles could then be readily ascertained due to the lower energy and therefore the smaller radius of curvature of the particles in the magnetic field after they had traversed the plate and suffered a loss in energy. Results were then obtained which could logically be interpreted only in terms of particles of a positive charge and a mass of the same order of magnitude as that normally possessed by the free negative electron. In paricular one photograph (see Fig. 1) shows a particle of positive charge traversing a 6 mm plate of lead. If electronic mass is assigned to this particle its energy before it traverses the plate is 63 million electron-volts and after itnemerges its energy is 23 million electron-volts. The possibility that this partitle of positive charge could represent a proton is ruled out on the basis of range and curvature.' back

David Hilbert (1925), On the Infinite, ' We encounter a completely different and quite unique conception of the notion of infinity in the important and fruitful method of ideal elements. The method of ideal elements is used even in elementary plane geometry. The points and straight lines of the plane originally are real, actually existent objects. One of the axioms that hold for them is the axiom of connection: one and only one straight line passes through two points. It follows from this axiom that two straight lines intersect at most at one point. There is no theorem that two straight lines always intersect at some point, however, for the two straight lines might well be parallel. Still we know that by introducing ideal elements, viz., infinitely long lines and points at infinity, we can make the theorem that two straight lines always intersect at one and only one point come out universally true. These ideal "infinite" elements have the advantage of making the system of connection laws as simple and perspicuous as possible. Another example of the use of ideal elements are the familiar complex-imaginary magnitudes of algebra which serve to simplify theorems about the existence and number of the roots of an equation.' back

Einstein, Podolsky & Rosen (1935), 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

Feynman, Leighton and Sands FLP II_02, Chapter 2: Differential Calculus of Vector Fields, ' Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. So we will spend much of our time analyzing them. In this way we will get a feel as to what should happen in different electromagnetic situations. On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential equations. They have the advantage of being fundamental and, so far as we know, precise. If you have learned the differential equations you can always go back to them. There is nothing to unlearn.' back

Feynman, Leighton and Sands FLP II_02, Chapter 2: Differential Calculus of Vector Fields, ' What it means really to understand an equation—that is, in more than a strictly mathematical sense—was described by Dirac. He said: “I understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it.” So if we have a way of knowing what should happen in given circumstances without actually solving the equations, then we “understand” the equations, as applied to these circumstances. A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist. ' back

Galileo Galilei (1633), Recantation of Galileo (June 22, 1633), ' Therefore, desiring to remove from the minds of your Eminences, and of all faithful Christians, this vehement suspicion, justly conceived against me, with sincere heart and unfeigned faith I abjure, curse, and detest the aforesaid errors and heresies, and generally every other error, heresy, and sect whatsoever contrary to the said Holy Church, and I swear that in the future I will never again say or assert, verbally or in writing, anything that might furnish occasion for a similar suspicion regarding me; ' back

John J. Hopfield (2024_12_08), Nobel Lecture: Physics is a point of view, ' John J. Hopfield delivered his Nobel Prize lecture "Physics is a point of view" on 8 December 2024 at the Aula Magna, Stockholm University. He was introduced by Professor Ellen Moons, Chair of the Nobel Committee for Physics. … back

Judith Bishop (2025_02_26), Erotica, gore and racism: how America’s war on ‘ideological bias’ is letting AI off the leash, ' Badly behaved artificial intelligence (AI) systems have a long history in science fiction. Way back in 1961, in the famous Astro Boy comics by Osamu Tezuka, a clone of a popular robot magician was reprogrammed into a super-powered thief. In the 1968 film 2001: A Space Odyssey, the shipboard computer HAL 9000 turns out to be more sinister than the astronauts on board think. More recently, real-world chatbots such as Microsoft’s Tay have shown that AI models “going bad” isn’t sci-fi any longer. Tay started spewing racist and sexually explicit texts within hours of its public release in 2016. The generative AI models we’ve been using since ChatGPT launched in November 2022 are generally well behaved. There are signs this may be about to change. On February 20, the US Federal Trade Commission announced an inquiry to understand “how consumers have been harmed […] by technology platforms that limit users’ ability to share their ideas or affiliations freely and openly”. Introducing the inquiry, the commission said platforms with internal processes to suppress unsafe content “may have violated the law”. ' back

Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, ' We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.' A Philosophical Essay on Probabilities, Essai philosophique dur les probabilites introduction to the second edition of Theorie analytique des probabilites based on a lecture given in 1794. back

Mary Sirridge (1999), Quam videndo intus dicimus: Seeing and Saying in De Trinitate XV, ' What is being asserted is that thought has the same form as seeing or speaking respectively, i.e., that it works essentially like seeing or speaking, that thought is a formal and functional isomorph of seeing or speaking.' back

Massimo Faggioli (2025_02_19), Donald Trump captured American Catholicism — and the ramifications are being felt around the world, ' American Catholicism is no longer just a refuge for socially conservative ideologues as it was between the 1990s and the early 2000s. It is now a brand, for sale to the highest bidder. Consider Vice President J.D. Vance, who arrived on the political scene thanks to masters of the universe such as Elon Musk and Peter Thiel, and converted to Catholicism on the road to Damascus of the new ressourcement Thomism — a Thomism and a ressourcement very different from that of the theologians of Vatican II like Yves Congar and Marie-Dominique Chenu. [. . .] Catholics for Trump, including those behind the “Project 2025”, are heirs of the patriarch of modern American Catholic conservatism, William F. Buckley, Jr., with his visions of society and race, of anti-Europeanism and isolationism, of the “procreative society”. Today these ideas are expressed in a crude and populist form, not elitist and refined against the masses. But there is nonetheless a genealogy of the recovery of anti-liberal Catholicism that goes back to Buckley and his followers, as well as to his nineteenth-century sources such as Edmund Burke’s reflections on the French Revolution. [. . .] And then there are the blatant lies that characterise much of the “Christian” propaganda for Trump since 2015. A façade of pro-Christian culture is used to pass off, misrepresent or justify virtually everything from his supposed pro-life beliefs, to his contempt for science, to his attacks on political opponents as “godless communists”. There are members of the Catholic hierarchy who have subjugated themselves to Trumpism and the goals of his campaign, going so far as to claim that former President Joe Biden was not a true Catholic — and in so doing so have embraced the manipulation of the word typical of this era. [. . .] In America, there is still a hunger for spirituality, community and God that is nurtured and sustained through politics. Trump’s populist movement is not only a response — albeit a simplistic, violent, vindictive one — to America’s economic and social uncertainties. It is also a response to that search for meaning that emerges from a world order that is visibly on the ropes. back

Meinard Kuhlmann (Stanford Encyclopedia of Philosophy), Quantum Field Theory, ' Quantum Field Theory (QFT) is the mathematical and conceptual framework for contemporary elementary particle physics. In a rather informal sense QFT is the extension of quantum mechanics (QM), dealing with particles, over to fields, i.e. systems with an infinite number of degrees of freedom. (See the entry on quantum mechanics.) In the last few years QFT has become a more widely discussed topic in philosophy of science, with questions ranging from methodology and semantics to ontology. QFT taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions of particles and fields, and even with some features of QM.' back

Michael Bordt (2011), Why Aristotle's God is not the Unmoved Mover, ' The aim of this essay is to show that the view—popular among certain philosophers and theologians—that Aristotle’s God is the unmoved mover is incorrect, or at least leads to serious misunderstanding. In a nutshell: among other things, the project of the twelfth book of the Metaphysics is to determine what the first ousia is. This first ousia is not identified with God in so far as it is an unmoved mover, but in so far as it is the actual activity (energeia) of thinking. To put matters differently, the actual activity of the first ousia does not consist in moving anything. Its activity rather consists in the exercise of reason, in thinking. Since, however, thinking is without qualification the best activity, and since God is that being who just does engage in the best activity, the first ousia, in so far as it is the same as the activity of thinking, must be God. Thus we perhaps expect that, at the summit of ontology, God himself will be the object of this first philosophy. Metaphysics Λ meets such an expectation only in a very limited way. The limitation is the following: that which, so to speak, stands at the summit of metaphysics is not God, but the activity of reason. While this activity is identified with God, it is not so identified directly or immediately, but only as mediated by way of the conception of the best possible life. The twelfth book of the Metaphysics thus provides to an even lesser extent than is usually assumed the outlines of a theology. By way of recompense, however, Aristotle offers us a truly breathtaking metaphysics.' back

Peter Osper (1957), Review: Albert Szent-Györgyi (1957): Bioenergetics, ' Everyone who is interested in biological chemistry will want to read and reread this book, and then design some experiments to prove Szent-Györgyi: right or wrong. One gets the impression that Szent-Györgyi will not be too unhappy to be proved wrong. . . .' In 1957 the scientist Albert Szent-Györgyi released this book which contained a part titled “Biological Structures and Functions”. The following statement without attribution was employed as an epigraph for this part (page 56): https://archive.org/details/bioenergetics00szen/page/57/mode/1up “Research is to see what everybody has seen and think what nobody has thought.” back

Rolf Landauer (1991), The physical nature of information, Abstract Information is inevitably tied to a physical representation and therefore to restrictions and possibilities related to the laws of physics and the parts available in the universe. Quantum mechanical superpositions of information bearing states can be used, and the real utility of that needs to be understood. Quantum parallelism in computation is one possibility and will be assessed pessimistically. The energy dissipation requirements of computation, of measurement and of the communications link are discussed. The insights gained from the analysis of computation has caused a reappraisal of the perceived wisdom in the other two fields. A concluding section speculates about the nature of the laws of physics, which are algorithms for the handling of information, and must be executable in our real physical universe.' back

Saint John Paul II (1992_10_31), x, ' However, the underlying problems of [the Galileo] case concern both the nature of science and the message of faith. It is therefore not to be excluded that one day we shall find ourselves in a similar situation, one which will require both sides to have an informed awareness of the field and of the limits of their own competencies. The approach provided by the theme of complexity could provide an illustration of this. 5. A twofold question is at the heart of the debate of which Galileo was the centre. The first is of the epistemological order and concerns biblical hermeneu­tics. In this regard, two points must again be raised. In the first place, like most of his adversaries, Galileo made no distinction between the scientific approach to natural phenomena and a reflection on nature, of the philosophical order, which that approach generally calls for. . . . Secondly, the geocentric representation of the world was commonly admit­ted in the culture of the time as fully agreeing with the teaching of the Bible, of which certain expressions, taken literally, seemed to affirm geocentrism. The problem posed by theologians of that age was, therefore, that of the compatibility between heliocentrism and Scripture. 6. From this we can now draw our first conclusion. The birth of a new way of approaching the study of natural phenomena demands a clarification on the part of all disciplines of knowledge. . . . 9. . . . The majority of theologians did not recognise the formal distinction between Sacred Scripture and its interpretation, and this led them unduly to transpose into the realm of the doctrine of the faith a question which in fact pertained to scientific investigation. . . . 12. . . .There exist two realms of knowledge, one which has its source in Revelation and one which reason can discover by its own power. To the latter belong especially the experimental sciences and philosophy. The distinction between the two realms of knowledge ought not to be understood as opposition. The two realms are not altogether foreign to each other; they have points of contact. The methodologies proper to each make it possible to bring out different aspects of reality. . . 14.  Humanity has before it two modes of development. The first involves culture, scientific research and technology, that is to say whatever falls within the horizontal aspect of man and creation, which is growing at an impressive rate. In order that this progress should not remain completely external to man, it presupposes a simultaneous raising of conscience, as well as its actuation. The second mode of development involves what is deepest in the human being, when, transcending the world and transcending himself, man turns to the One who is the Creator of all. It is only this vertical direction which can give full meaning to man’s being and action, because it situates him in relation to his origin and his end. In this twofold direction, horizontal and vertical, man realises himself fully as a spiritual being and as homo sapiens. But we see that development is not uniform and linear, and that progress is not always well ordered. This reveals the disorder which affects the human condition. The scientist who is conscious of this twofold development and takes it into account contributes to the restoration of harmony. Those who engage in scientific and technological research admit, as the premise of its progress, that the world is not a chaos but a ‘cosmos’; that is to say, that there exist order and natural laws which can be grasped and examined, and which, for this reason, have a certain affinity with the spirit. Einstein used to say: ‘What is eternally incomprehensible in the world is that it is comprehensible’.9 This intelligibility, attested to by the marvellous discoveries of science and technology, leads us, in the last analysis, to that transcendent and primordial Thought imprinted on all things.' back

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

W. F. McGrew et al, Atomic clock performance enabling geodesy below the centimetre level, ' The passage of time is tracked by counting oscillations of a frequency reference, such as Earth’s revolutions or swings of a pendulum. By referencing atomic transitions, frequency (and thus time) can be measured more precisely than any other physical quantity, with the current generation of optical atomic clocks reporting fractional performance below the 10−17 level. However, the theory of relativity prescribes that the passage of time is not absolute, but is affected by an observer’s reference frame. Consequently, clock measurements exhibit sensitivity to relative velocity, acceleration and gravity potential. Here we demonstrate local optical clock measurements that surpass the current ability to account for the gravitational distortion of space-time across the surface of Earth. In two independent ytterbium optical lattice clocks, we demonstrate unprecedented values of three fundamental benchmarks of clock performance. In units of the clock frequency, we report systematic uncertainty of 1.4 × 10−18, measurement instability of 3.2 × 10−19 and reproducibility characterized by ten blinded frequency comparisons, yielding a frequency difference of [−7 ± (5)stat ± (8)sys] × 10−19, where ‘stat’ and ‘sys’ indicate statistical and systematic uncertainty, respectively. Although sensitivity to differences in gravity potential could degrade the performance of the clocks as terrestrial standards of time, this same sensitivity can be used as a very sensitive probe of geopotential. Near the surface of Earth, clock comparisons at the 1 × 10−18 level provide a resolution of one centimetre along the direction of gravity, so the performance of these clocks should enable geodesy beyond the state-of-the-art level. These optical clocks could further be used to explore geophysical phenomena, detect gravitational waves, test general relativity and search for dark matter.' back

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