This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 9: The active creation of Hilbert space
Table of contents
9.0: Introductory note
9.1: Continuous creation
9.2: Formalism
9.3: Hilbert space
9.4: The creation of the Hilbert space of the universe
9.5: Hilbert space is invisible
9.0: Introductory note
The starting point for this site was the Roman Catholic religion that I experienced in my youth. This religion, as we know it, was created by the warlord Constantine the Great when he assembled the Christian Bishops at Nicea. The bishops produced the Nicene Creed which served as the text of a new religious foundation for Constantine's empire. Nicene Creed - Wikipedia, Constantine the Great and Christianity - Wikipedia
From the state's point of view, the time and effort spent on theological debate was a small price to pay for symbolic unification. The donation of sizeable gifts to the church from private and public funds was part, too, of that gradual unification. Besides, state generosity and popular piety helped divert attention away from the vast profits which the Roman state extracted by monetizing the huge stored treasured of pagan official religion, accumulated over centuries of piety and peace. The change of official religion from paganism to Christianity created enormous windfalls for the Roman state. Keith Hopkins (2001): A World Full of Gods: The Strange Triumph of Christianity page 105
This project shares Constantine's motivation. He wanted a spiritual, religious and theological ground for peace in his Empire. My feeling is that a scientific evidence based theology may show us the way global peace by exploring the way our amazing world came to be. The physical foundation of this peace must be protection of our common habitat and fair distribution of capital and income. My model is the harmony existing in my own body established by the genome shared by all my cells. These trillions of individuals work together as a harmonious whole, feeding and protecting one another in return for services rendered to my whole being. In an application of this model our common "genome" is a shared scientific theology, an ideology or theory of everything based on the realities of our environment. As scientific biology has unified health care around the world, we might hope that scientific theology will eventually unite all of us humans in a common spiritual vision of our role in the divine world. As Jesus of Nazareth put it, love god, love your neighbour is the whole of the law (Matthew 22: 36–40). The associated Parable of the Good Samaritan teaches us to respect and care for one another (Luke, 10: 25–37). At present national ideologies often override this law, leading to widespread military rape, pillage and murder. Pope Francis (2015): Laudato Si': On care of our common home, Thomas Piketty (2020): Capital and Ideology, Piketty (2022): A Brief History of Equality,
Roman Christianity is framed as a battle between good and evil. The Hebrew Bible records the initial struggle between Yahweh and Satan. The Christian New Testament documents the final victory of the Triune God. God the Father was satisfied for the Original Sin committed by the first humans under the influence of Satan by the murder of the human incarnation of his Son, Jesus of Nazareth. By this human sacrifice the Church believes that the world was redeemed from the power of Satan. The damage done to it by the Father as punishment of all the children of the first people is eventually to be repaired at the end of the world. The Holy Spirit is to guide us towards this final salvation.The Book of Genesis 3: The Fall and God's Punishment of Humanity, Salvation in Christianity - Wikipedia
The previous pages of this site describe the current theological foundations of Christianity. I have preserved all the elements of this theological story that may serve as foundations for science. From this point on I reject the idea that evil is the result of sin caused by malign spirits like Satan. Instead such evil as there is in the world seems to be consequence of evolution. Since this is a natural process it can be scientifically understood and managed by population control, equitable sharing of resources and giving democratic control to everybody affected by collective enforceable decisions.
9.1: Continuous creation
Before it was replaced by the big bang model, the continuous creation or steady state theory of the Universe was popular. As the Universe expands, the theory goes, new matter is created to fill the new space. Locally, the Universe has always been the same, and may be eternal as many of the ancients thought. Steady-state model - Wikipedia, Observable Universe - Wikipedia
The evolutionary big bang model attributes a beginning to Universe. The idea is that the Universe began from a struturelesss initial singularity which contained all the energy of the Universe and at some point exploded in a big bang to create the Universe as we know it. The Universe has evolved to its current size and structure in the 14 billion years since that event. The Universe, measured in energy, has become less dense as its volume increases. At the moment it averages about 5 hydrogen atoms per cubic metre, less dense than the best vacuum we can create on Earth. Initial singularity - Wikipedia, Big Bang - Wikipedia
Here we are seeking to model the Universe as the physical mind of god. The language we are using to describe it is a cross between the ancient theological discussion of divinities and angels and the modern understanding of quantum theory as a description of the communication and associated computation between particulate sources. Quantum theory works at all scales from the quantum of action to the Universe as a whole. The creation of mental and cultural space is unbounded, reflected in Cantor's transfinite numbers and the exceedingly complex structures around us like the living Earth and ourselves. Nielsen & Chuang (2016): Quantum Computation and Quantum Information, Transfinite numbers - Wikipedia, Jeffrey Nicholls (1987): A theory of Peace
9.2: Formalism
There are two difficulties with the big bang story. First, where did the initial singularity, which is considered to be prior to space and time, come from? We can say that it is eternal, like the traditional divinity, since time does not exist for it. The second problem is that we associate energy and momentum with space and time, so we may wonder how all the energy of the Universe could have been contained in an entity existing prior to space and time.
An alternative approach may be to consider the initial singularity as a logical or psychological quantum object analogous to Aristotle's unmoved mover and Aquinas's model of the Christian God, an entity of pure actuality, actus purus, completely simple, omnino simplex, but not in initially quantized. We shall call this singularity naked gravitation. Later, beginning on page 16: Potential + kinetic = zero energy universe and continued on page 17: Gravitation and quantum theory—in the beginning we shall explore the idea that gravitational potential supplies the energy necessary to convert the formal kinematic structures of Hilbert space and quantum theory into dynamic physical realities, real particles. These particles, by their interactions, create the Minkowski space described by special relativity. This is the foundation of the Einstein space described by general relativity. Unmoved mover - Wikipedia, Aquinas, Summa: I, 2, 3: Does God exist?
The Christian idea of God developed in the New Tesament is descended from the God of the Hebrew Bible. The Hebrew Bible was taken over by Christianity, rearranged, edited and renamed the Old Testament. The Christian New Testament, developed in the centuries after the death of Jesus, records two major developments of the Hebrew God. First, they appeared on Earth as a human person, Jesus of Nazareth. As John's Gospel records, the immaterial and immortal Word of God became human and lived among us. Gospel, John 1: And the Word was made flesh
A parallel development was the introduction of the Trinity. Jesus claimed to be the Son of God, which implied that he had a divine father, God the Father, who appears to be the same person as Yahweh of the Hebrew Bible. We also find in the New Testament references to the Holy Spirit, also a divine person. The theory of the Trinity developed by Augustine, Aquinas and Lonergan holds that the second two persons proceeded from the Father (page 8: The theology of the Trinity). The emergence of the Trinity implies divinity can proceed from divinity. Although Christian doctrine limits this process to three persons, we see no logical reason why the process cannot go further.
It may feel counterintuitive to identify the quantum of action, the smallest entity in the Universe, with the divinity, the largest entity imaginable. The formal approach makes this identification possible by abstracting from all concrete size and detail and standing upon a simple definition alone: an act is an event that changes the world just as a divinity is an event that changes the world. A quantum of action, like a divinity, is a discrete agent limited in its omipotence by consistency alone. Aquinas, Summa I, 25, 3: Is God omnipotent?
In its simplest incarnation, we may consider the quantum of action as a not operator, an activity which may annihilate something existing and create something new. This operator changes some situation p into some situation not-p. In the binary realm of propositional logic, we understand that not-p annihilates p and creates not-p. On the other hand, not-not-p ≡ p reverses this operation, annihilating non-p and recreating p.
We may think of this operation in a physical way as a pendulum. A pendulum oscillates between kinetic and potential energy. At the top of its swing, the bob has gravitational potential; as it swings down this potential energy is converted into kinetic energy. On the upswing, the opposite happens. In an ideal frictionless world, a pendulum would swing forever. Then we think of it in formal mathematical terms as an harmonic oscillator.
In the wider world of space, as in the Trinity, new things can be created without destroying the old. The Father remains in existence as the Son proceeds from them. The Father and the Son remain in existence as the Holy Spirit proceeds, and we can imagine constructing a Universe by an unbounded sequence of processions. Each new person or source is a new quantum of action, a new creator. Action, as we conceive it here, exists before space and time, just like the traditional divinity and the initial singularity. Like God, it is hard to imagine. It is not a thing but an event, something inherently in action, in no way confined to any particular action but unable, like God, to create a contradiction.
Our interpretation of action in terms of logic may be understood to be purely formal. Logic, like formal mathematics, is kinetic. A puppet without agency. Hilbert, Whitehead and Russell's formal mathematics may be considered to exist outside spacetime. Nevertheless it does have spacetime representation in the mathematical literature and is moved kinematically by the minds (brains) of mathematicians, students and the Universe as a whole. Wigner's observation on the utility of mathematics in science also suggests that is is embodied in reality. Hilbert's program - Wikipedia, Whitehead & Russell (1910, 1962): Principia Mathematica, Eugene Wigner (1960): The Unreasonable Effectiveness of Mathematics in the Natural Sciences
9.3: Hilbert space
Quantum mechanics began with Max Planck's realization that quantization prevented the ultraviolet catastrophe associated with treating black body radiation as a continuum. It took another thirty years to reach its final form. In 1930 Dirac published a treatise which was to become the standard reference, based on his transformation theory. John von Neumann then perfected the mathematical presentation of Dirac's work, locating it in an abstract Hilbert space. Black-body radiation - Wikipedia, Ultraviolet catastrophe - Wikipedia, Paul Dirac (1983): The Principles of Quantum Mechanics (4th ed)
Hilbert space is a complex linear vector space analogous to Cartesian space, with a metric defined by an inner product. Physical states are represented by rays in this Hilbert space. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics, Inner product space - Wikipedia
Von Neumann defines an abstract n dimensional dimensional Hilbert space (C n) with three axioms.
α) A “scalar product,” i.e., the product of a (complex) number a with an element f of Hilbert space: af;
β) Addition and subtraction of two elements f, g of Hilbert space: f ± g;
γ) An “inner product” of two elements f, g in Hilbert space. Unlike α and β this operation produces a complex number which is not an element of Hilbert space: (f, g) but a metric, a measure of "distance".
The theory of quantum computation and quantum information operates in n dimensional Hilbert pace. Nielsen & Chuang (2016): Quantum Computation and Quantum Information page xxix
Each element, f, g, . . . defines the orientation of a complex plane in Hilbert space. It is called a ray, with the property that e i θ f = f so that the orientation of an element within its associated plane is only relevant when vectors associated with other planes are added (superposed).
Basis elements of a Hilbert space f and g are normalized by the property (f, f) = 1 and are said to be orthogonal when (f, g) = 0.
Each vector in a Hilbert space may represent a quantum of action, and we assume that since the specific dynamic property of a quantum of action is to act, the initial singularity will eventually become populated with a ℵ0 states. The final result is an ℵ0 dimensional Hilbert space of orthonormal basis vectors. On page 23: Insight and belief we relate this structure to modern field theory. Superpositions (ie vector additions) of these basis vectors are normalized to represent quantum states.
Von Neuman points out that:
The noteworthy feature of the operations af, f ± g, (f, g) is that they are exactly the basic operations of the vector calculus: those which make possible the establishment of length and angle calculations in Euclidean geometry or the calculations with force and work in the mechanics of particles.
Here we find another interesting feature of ancient philosophy. Each of Plato's forms is unique. Aristotle, when he brought the forms down to Earth, enabled the existence of multiple individuals of each species by embodying the forms in matter. Matter thus became known as the principle of individuation. The old theologians faced a similar problem with the multiplicity of angels. Since angels are held to be immaterial there can be no material principle of individuation. There can therefore be only one angel in each species, since it was believed that formal differences are needed to differentiate immaterial structures. We might imagine that vectors in Hilbert states would be useful for naming angels. They re differentiated by their orthogonality. All their pairwise inner products are zero. Principle of individuation - Wikipedia, Aquinas, Summa I, 50, 4: Is every angel a different species?
The abstract Hilbert space defined by von Neumann as the domain of quantum mechanics is a mathematically ideal formal structure. Here we imagine the ultimate creation of a series of Hilbert spaces with dimension ranging from 1 to ℵ0 which constitute the formal foundation of a quantum mechanical vacuum. The orthogonality of this set of basis vectors is guaranteed by the quantum no cloning theorem. All Hilbert spaces are essentially identical, differentiated only by their number of dimensions. We may say that they are symmetrical with respect to complexity. Hilbert space - Wikipedia, No-cloning theorem - Wikipedia
To these axioms for finite Hilbert space, von Neuman adds the notions of completeness and separability to describe infinite dimensional (C ∞) space. Complete metric space - Wikipedia, Separable space - Wikipedia
9.4: The creation of the Hilbert space of the Universe
We model the initial singularity, which we may think of as analogous to the traditional God, following the work of Aristotle and Aquinas, as pure act, actus purus. We may think of it as like God, infinite, that is not bounded by or contained by anything, visually like an infinite cloud. There is nevertheless some constraint on the power of God: they are bound by consistency. They cannot create a contradiction. Aquinas, Summa: I, 7, 1: Is God infinite?
The theological model of the Trinity is based on two ideas, the procession of the divine persons and their distinction. Augustine, Aquinas and Lonergan all based the procession of the Son and the Spirit on the conception and expression of an idea in the human mind. Although the contents of our minds are consdered to be accidental, they applied the principle that is what is accidental in created beings is necessarily substantial in God, since it is axiomatic that in God there can be no distinction between substance and accidents. Aquinas, Summa I, 28, 1: Are there real relations in God?
All three of these authors base the distinction of the persons on the idea of relationship, relying on the fact that the persons are distinguished by their relationships to the source from which they proceed, the Son to the Father and the Spirit to the Son and the Father together. Here we rely on the defintion of Hilbert space rather than relationships. All the basis states (dimensions) of Hilbert space are understood to be orthogonal - their pairwise inner products are zero.
We can now apply these ideas to the creation of the Hilbert space defined by von Neumann.
We assume that the initial singularity is pure action identical to the traditional God, and that it has the power to reproduce itself indefinitely free of the dogmatic limitations derived from the interpretation of the Trinity in the New Testament. We assume that a sequence of orthogonal actions creates Hilbert space, dimension by dimension.
We may identify the simplest Hilbert space with the initial singularity, a zero dimensional Hilbert space which we may represent by the symbol |0⟩. We might imagine it to be formally identical to the empty set ∅, with no internal content or structure. This is consistent with the traditional idea that God is absolutely simple, omnino simplex. We can represent the first new vector created by the action of the initial singularity by the symbol |1⟩. This vector is orthogonal to |0⟩, that is distinct from it, so a qubit is a 2D space within the initial singularity, as the persons of the Trinity are within God. The superposition of |0⟩ and |1⟩ gives a qubit, represented by the the symbol |qubit⟩ = a|0⟩ + b|1⟩, where a and b are complex numbers. Qubit - Wikipedia
The qubit and all subsequent superpositions of newly created vectors a|0⟩ + b|1⟩ + c|2⟩ . . . are normalized by the requirement that a2 + b2 + c2 . . . = 1. This normalization, which is maintained by the unitary operators of quantum systems, establishes that any quantum system obeys the probabilistic structure of a communication source, to be described on page 11: Quantization: the mathematical theory of communication. Normalization, Unitary operator - Wikipedia
Because all the quantum basis states |0⟩, |1⟩, |2⟩ . . . are orthogonal and we intend to use the primordial Hilbert space as the computational basis of the Universe, we work in Cn where the ℵ0 is the upper bound of n equinumerous with the set of natural numbers and the set of Turing machines or computable functions. On the polar complex plane the basis states of a Hilbert space may be thought of as points on a circle with unit radius r representing the absolute value of each state and a rational argument θ defining the position of the point on the circle. Complex number - Wikipedia, J D Cresser (2009): General Mathematical Description of a Quantum System
We may imagine the creation of quanta of action within the initial singularity as a consequence of the mathematical theory of fixed points. We may assume that since the initial singularity is all that there is is may be understood as a continuous, convex bounded set and therefore subject to theorems like Brouwer's fixed point theorem. We may imagine that fixed point theorems yield different fixed points depending on the function used to map set onto itself. The ℵ0 fixed points that we may find in the initial singularity and its children may therefore be correlated with the ℵ0 Turing computable functions. Brouwer fixed point theorem - Wikipedia, Computable function - Wikipedia
9.5: Hilbert space is invisible
The traditional God is invisible, and this is also true of the quantum operations that underlie the observable features of the Universe. This is because in a quantum Universe to observe and to be observed both require action. We will discuss this in more detail on page 14: Measurement: the interface between Hilbert and Minkowski and page 15: Quantum amplitudes and logical processes are invisible. This situation occurs at all scales. In a nutshell, if somebody asks you what you are doing when you are involved in an engrossing task, you usually have to stop what you are doing to explain yourself.
This idea assumes that fundamental processes of the Universe are fully taken up with their own work and cannot spare the resources to explain themselves without leaving their task. I feel that the fundamental processes of the Universe are computable functions executed by quantum computations whose operation is invisible because their essential roles cannot be interrupted.
Here we come to the role of evolution and creation, differentiating the strucureless initial singularity into the structured Universe by the ceaseless emission of quanta of action within the initial singularity. This process is analogous to the traditional divine operation of creating and sustaining the Universe. The medium of this interaction is the flow of quanta of action which we interpret as energy using the Planck-Einstein relation E - hf. This, we might understand, is a possible quantum rendering of the big bang. Planck-Einstein relation - Wikipedia
(revised Tuesday 6 August 2024)
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Notes and references
Further reading
Books
Dirac (1983), P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)
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Hopkins (2001), Keith, A World Full of Gods: The Strange Triumph of Christianity, Penguin Random House 2001 ' In this provocative, irresistibly entertaining book, Keith Hopkins takes readers back in time to explore the roots of Christianity in ancient Rome. Combining exacting scholarship with dazzling invention, Hopkins challenges our perceptions about religion, the historical Jesus, and the way history is written. He puts us in touch with what he calls “empathetic wonder”—imagining what Romans, pagans, Jews, and Christians thought, felt, experienced, and believed-by employing a series of engaging literary devices. These include a TV drama about the Dead Sea Scrolls; the first-person testimony of a pair of time-travelers to Pompeii; a meditation on Jesus’ apocryphal twin brother; and an unusual letter on God, demons, and angels.'
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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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Piketty (2020), Thomas, Capital and Ideology, Harvard University Press 2020 x politics of identity.
Once we understand this, we can begin to envision a more balanced approach to economics and politics. Piketty argues for a new "participatory" socialism, a system founded on an ideology of equality, social property, education, and the sharing of knowledge and power. Capital and Ideology is destined to be one of the indispensable books of our time, a work that will not only help us understand the world, but that will change it.
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Piketty (2022), Thomas, A Brief History of Equality, Harvard UP 2022 ' The world's leading economist of inequality presents a short but sweeping and surprisingly optimistic history of human progress toward equality despite crises, disasters, and backsliding. A perfect introduction to the ideas developed in his monumental earlier books.It's easy to be pessimistic about inequality. We know it has increased dramatically in many parts of the world over the past two generations.'
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Whitehead (1910, 1962), Alfred North, and Bertrand Arthur Russell, Principia Mathematica (Cambridge Mathematical Library), Cambridge University Press 1910, 1962 The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. Not long after it was published, Gödel showed that the project could not completely succeed, but that in any system, such as arithmetic, there were true propositions that could not be proved.
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Links
Aquinas, Summa I, 25, 3, Is God omnipotent?, '. . . God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.' back |
Aquinas, Summa I, 28, 1, Are there real relations in God?, 'Reply to Objection 4. Relations which result from the mental operation alone in the objects understood are logical relations only, inasmuch as reason observes them as existing between two objects perceived by the mind. Those relations, however, which follow the operation of the intellect, and which exist between the word intellectually proceeding and the source whence it proceeds, are not logical relations only, but are real relations; inasmuch as the intellect and the reason are real things, and are really related to that which proceeds from them intelligibly; as a corporeal thing is related to that which proceeds from it corporeally. Thus paternity and filiation are real relations in God.' back |
Aquinas, Summa I, 50, 4, Is every angel a different species?, ' . . . such things as agree in species but differ in number, agree in form, but are distinguished materially. If, therefore, the angels be not composed of matter and form, as was said above (Article 2), it follows that it is impossible for two angels to be of one species; just as it would be impossible for there to be several whitenesses apart, or several humanities, since whitenesses are not several, except in so far as they are in several substances.' back |
Aquinas, Summa: I, 2, 3, Does God exist?, 'I answer that, The existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . ' back |
Aquinas, Summa: I, 7, 1, Is God infinite?, ' I answer that, All the ancient philosophers attribute infinitude to the first principle, as is said (Phys. iii), and with reason; for they considered that things flow forth infinitely from the first principle. . . . .. Now being is the most formal of all things, as appears from what is shown above (I:4:1 Objection 3). Since therefore the divine being is not a being received in anything, but He is His own subsistent being as was shown above (I:3:4), it is clear that God Himself is infinite and perfect.' back |
Big Bang - Wikipedia, Big Bang - Wikipedia, the free encyclopedia, ' The Big Bang theory is the prevailing cosmological model explaining the existence of the observable universe from the earliest known periods through its subsequent large-scale evolution. The model describes how the universe expanded from an initial state of high density and temperature, and offers a comprehensive explanation for a broad range of observed phenomena, including the abundance of light elements, the cosmic microwave background (CMB) radiation, and large-scale structure. ' back |
Black-body radiation - Wikipedia, Black-body radiation - Wikipedia, the free encyclopedia, 'Black-body radiation is the type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body) held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body.' 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 |
Complete metric space - Wikipedia, Complete metric space - Wikipedia, the free encyclopedia, ' In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. √2 is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, . . . ' back |
Complex number - Wikipedia, Complex number - Wikipedia, the free encyclopedia, 'A complex number is a number that can be expressed in the form a + bi, where a. and b are real numbers and |
Computable function - Wikipedia, Computable function - Wikipedia, the free encyclopedia, 'Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions.' back |
Constantine the Great and Christianity - Wikipedia, Constantine the Great and Christianity - Wikipedia, the free encyclopedia, ' During the reign of the Roman Emperor Constantine the Great (AD 306–337), Christianity began to transition to the dominant religion of the Roman Empire. Historians remain uncertain about Constantine's reasons for favoring Christianity, and theologians and historians have often argued about which form of early Christianity he subscribed to. . . . Constantine's decision to cease the persecution of Christians in the Roman Empire was a turning point for early Christianity, sometimes referred to as the Triumph of the Church, the Peace of the Church or the Constantinian shift. In 313, Constantine and Licinius issued the Edict of Milan decriminalizing Christian worship. The emperor became a great patron of the Church and set a precedent for the position of the Christian emperor within the Church and raised the notions of orthodoxy, Christendom, ecumenical councils, and the state church of the Roman Empire declared by edict in 380. He is revered as a saint and is apostolos in the Eastern Orthodox Church, Oriental Orthodox Church, and various Eastern Catholic Churches for his example as a "Christian monarch”.' back |
Eugene Wigner (1960), The Unreasonable Effectiveness of Mathematics in the Natural Sciences, ' The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back |
Gospel, John 1, And the Word was made flesh , ' 14: And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth. 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 a vector space equipped with an inner product, an operation that allows defining lengths and angles. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.' back |
Hilbert's program - Wikipedia, Hilbert's program - Wikipedia, the free encyclopedia, ' In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.' back |
Initial singularity - Wikipedia, Initial singularity - Wikipedia, the free encyclopedia, ' The initial singularity is a singularity predicted by some models of the Big Bang theory to have existed before the Big Bang. The instant immediately following the initial singularity is part of the Planck epoch, the earliest period of time in the history of our universe. The use of only general relativity to predict what happened in the beginnings of the Universe has been heavily criticized, as quantum mechanics becomes a significant factor in the high-energy environment of the earliest Universe, and general relativity on its own fails to make accurate predictions. Although there is no direct evidence for a singularity of infinite density, the cosmic microwave background is evidence that the universe expanded from a very hot, dense state. ' back |
Inner product space - Wikipedia, Inner product space - Wikipedia, the free encyclopedia, 'In mathematics, an inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry. The additional structure associates to each pair of vectors in the space a number which is called the inner product (also called a scalar product) of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the angle between vectors or length of vectors in spaces of all dimensions. It also allows introduction of the concept of orthogonality between vectors. Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis. An inner product space is sometimes also called a pre-Hilbert space, since its completion with respect to the metric induced by its inner product is a Hilbert space.' back |
J D Cresser (2009), General Mathematical Description of a Quantum System, ' This idea that the state of a quantum system is to be considered a vector belonging to a complex vector space, which we have developed here in the case of a spin half system, and which has its roots in the sum over paths point of view, is the basis of all of modern quantum mechanics and is used to describe any quantum mechanical system. Below is a summary of the main points as they are used for a general quantum system whose state spaces are of arbitrary dimension (including state spaces of infinite dimension). The emphasishere is on the mathematical features of the theory.' back |
Jeffrey Nicholls (1987), A theory of Peace, ' The argument: I began to think about peace in a very practical way during the Viet Nam war. I was the right age to be called up. I was exempted because I was a clergyman, but despite the terrors that war held for me, I think I might have gone. It was my first whiff of the force of patriotism. To my amazement, it was strong enough to make even me face death. |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Nicene Creed - Wikipedia, Nicene Creed - Wikipedia, the free encyclopedia, ' The Nicene Creed (Greek: Σύμβολον τῆς Νίκαιας, Latin: Symbolum Nicaenum) is the profession of faith or creed that is most widely used in Christian liturgy. It forms the mainstream definition of Christianity for most Christians. It is called Nicene because, in its original form, it was adopted in the city of Nicaea (present day Iznik in Turkey) by the first ecumenical council, which met there in the year 325. The Nicene Creed has been normative for the Catholic Church, the Eastern Orthodox Church, the Church of the East, the Oriental Orthodox churches, the Anglican Communion, and the great majority of Protestant denominations.' back |
No-cloning theorem - Wikipedia, No-cloning theorem - Wikipedia, the free encyclopedia, ' In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist . . .. The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors.' back |
Normalization, Wave function; Normalization condition, ' For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details).' back |
Observable Universe - Wikipedia, Observable Universe - Wikipedia, the free encyclopedia, 'The observable universe is a spherical region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because electromagnetic radiation from these objects has had time to reach the Solar System and Earth since the beginning of the cosmological expansion. There are at least 2 trillion galaxies in the observable universe.' back |
Planck-Einstein relation - Wikipedia, Planck-Einstein relation - Wikipedia, the free encyclopedia, 'The Planck–Einstein relation. . . refers to a formula integral to quantum mechanics, which states that the energy of a photon (E) is proportional to its frequency (ν). E = hν. The constant of proportionality, h, is known as the Planck constant.' back |
Pope Francis (2015), Laudato Si': On care of our common home, '1. “LAUDATO SI’, mi’ Signore” – “Praise be to you, my Lord”. In the words of this beautiful canticle, Saint Francis of Assisi reminds us that our common home is like a sister with whom we share our life and a beautiful mother who opens her arms to embrace us. “Praise be to you, my Lord, through our Sister, Mother Earth, who sustains and governs us, and who produces various fruit with coloured flowers and herbs”. 2. This sister now cries out to us because of the harm we have inflicted on her by our irresponsible use and abuse of the goods with which God has endowed her. We have come to see ourselves as her lords and masters, entitled to plunder her at will. The violence present in our hearts, wounded by sin, is also reflected in the symptoms of sickness evident in the soil, in the water, in the air and in all forms of life. This is why the earth herself, burdened and laid waste, is among the most abandoned and maltreated of our poor; she “groans in travail” (Rom 8:22). We have forgotten that we ourselves are dust of the earth (cf. Gen 2:7); our very bodies are made up of her elements, we breathe her air and we receive life and refreshment from her waters.' back |
Principle of individuation - Wikipedia, Principle of individuation - Wikipedia, the free encyclopedia, ' The principle of individuation is a criterion that individuates or numerically distinguishes the members of the kind for which it is given, that is by which we can supposedly determine, regarding any kind of thing, when we have more than one of them or not. It is also known as a 'criterion of identity' or 'indiscernibility principle'. The history of the consideration of such a principle begins with Aristotle. It was much discussed by the medieval philosopher Duns Scotus (c. 1266–1308) with his "haecceity" and later, during Renaissance, by Francisco Suárez (1548–1617), Bonaventure Baron (1610–1696) and Leibniz (1646–1716). back |
Quantum computing - Wikipedia, Quantum computing - Wikipedia, the free encyclopedia, ' Quantum computing began in the early 1980s, when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine. Richard Feynman and Yuri Manin later suggested that a quantum computer had the potential to simulate things that a classical computer could not. In 1994, Peter Shor developed a quantum algorithm for factoring integers that had the potential to decrypt RSA-encrypted communications. Despite ongoing experimental progress since the late 1990s, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream." ' back |
Qubit - Wikipedia, Qubit - Wikipedia, the free encyclopedia, 'A quantum bit, or qubit . . . is a unit of quantum information. That information is described by a state vector in a two-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers. Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. In the acknowledgments of his paper (Phys. Rev. A 51, 2738), Schumacher states that the term qubit was invented in jest, during his conversations with Bill Wootters.' back |
Salvation in Christianity - Wikipedia, Salvation in Christianity - Wikipedia, the free encyclopedia, ' In Christianity, salvation (also called deliverance or redemption) is the "saving [of] human beings from sin and its consequences, which include death and separation from God" by Christ's death and resurrection, and the justification following this salvation. While the idea of Jesus' death as an atonement for human sin was recorded in the Christian Bible, and was elaborated in Paul's epistles and in the Gospels, Paul saw the faithful redeemed by participation in Jesus' death and rising. Early Christians regarded themselves as partaking in a new covenant with God, open to both Jews and Gentiles, through the sacrificial death and subsequent exaltation of Jesus Christ.' back |
Separable space - Wikipedia, Separable space - Wikipedia, the free encyclopedia, 'In mathematics a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence . . . of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case;) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.' back |
Steady-state model - Wikipedia, Steady-state model - Wikipedia, the free encyclopedia, ' In cosmology, the steady-state model is an alternative to the Big Bang theory of evolution of the universe. In the steady-state model, the density of matter in the expanding universe remains unchanged due to a continuous creation of matter, thus adhering to the perfect cosmological principle, a principle that asserts that the observable universe is practically the same at any time and any place.' back |
The Book of Genesis 3, The Fall and God's Punishment of Humanity, '1. Now the snake was the most cunning of all the wild animals that the LORD God had made. He asked the woman, “Did God really say, ‘You shall not eat from any of the trees in the garden’?” |
Transfinite numbers - Wikipedia, Transfinite numbers - Wikipedia, the free encyclopedia, 'Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary workers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.' back |
Ultraviolet catastrophe - Wikipedia, Ultraviolet catastrophe - Wikipedia, the free encyclopedia, ' The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 statistical derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh–Jeans law accurately predicts experimental results at radiative frequencies below 1055 GHz, but begins to diverge with empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum. Since the first appearance of the term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.' back |
Unitary operator - Wikipedia, Unitary operator - Wikipedia, the free encyclopedia, ' In functional analysis, a branch of mathematics, a unitary operator . . . is a bounded linear operator U : H → H on a Hilbert space H satisfying UU* = U*U = I where U* is the adjoint of U, and I : H → H is the identity operator. This property is equivalent to the following:
1. U preserves the inner product ( , ) of the Hilbert space, ie for all vectors x and y in the Hilbert space, (Ux, Uy) = (x, y) and |
Unmoved mover - Wikipedia, Unmoved mover - Wikipedia, the free encyclopedia, ' The unmoved mover (Ancient Greek: ὃ οὐ κινούμενον κινεῖ, lit. 'that which moves without being moved' or prime mover (Latin: primum movens) is a concept advanced by Aristotle as a primary cause (or first uncaused cause) or "mover" of all the motion in the universe. As is implicit in the name, the unmoved mover moves other things, but is not itself moved by any prior action. In Book 12 (Greek: Λ) of his Metaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfect contemplation: self-contemplation. He equates this concept also with the active intellect. This Aristotelian concept had its roots in cosmological speculations of the earliest Greek pre-Socratic philosophers and became highly influential and widely drawn upon in medieval philosophy and theology. St. Thomas Aquinas, for example, elaborated on the unmoved mover in the Quinque viae. ' back |