Commentary on Home Page Step 2: The generation of random abstract Hilbert space
Next we imagine that Brouwer’s fixed point theorem maps the Aquinas–Einstein symmetry onto itself establishing the axiomatic basis for the abstract Hilbert space described by von Neumann. This will be the foundation for the quantum mechanics emerging in Step 3. Since the singularity is structureless and therefore has no means to control itself its actions are random, so providing the variation necessary for creative evolution. Brouwer fixed point theorem - Wikipedia, John L. Casti (1996): Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John von Neumann (2018): The Mathematical Foundations of Quantum Mechanics
Table of Contents
1. Introduction: Hilbert space is the foundation of the Universe
2. How do we imagine a quantum of action
3. Von Neumann’s axioms for abstract Hilbert Space.
4. This Hilbert space lies beneath the Minkowski space in which we live
5. The logical completeness of Hilbert space that renders the universe divine
1. Introduction: Hilbert space is the foundation of the Universe
We begin our construction of the universe with an initial, eternal, structureless entity we have called the Aquinas–Einstein symmetry. The simplest element of the world is the quantum of action and we may think of this initial symmetry as the primordial quantum of action. It is an analogue of the god in the traditional Catholic theology which Aquinas defines as actus purus, pure activity. Muslim theologians like Ibn Sina aka Avicenna (c 980 – 1037) thought of this entity as the necessary being, a being whose essence is to exist. In many traditional theologies, the primordial being is understood to be invisible to us, creating and managing the universe we inhabit from the outside. Here we see the primordial quantum of action creating our universe within itself by the process of random variation and selection selected by Charles Darwin and others to explain the origin of life. Alexis Szejnoga (2013): Persian Perspective on Prima Philosophia: The influence of Avicenns's interpetation of Aristotelian ontology in the De ente et essentia of Saint Thomas Aquinas
Randomness is necessary because it seems that deterministic processes like the work of Laplace’s Demon are doomed to endlessly repeat the past and cannot be creative. We conceive the notion of a random event through mathematical ideas like Kolmogorov’s theory of probability. Kolmogorov
set himself the task [in 1933] of putting in their natural place, among the general notions of modern mathematics, the basic concepts of probability theory—concepts until quite recently were considered to be quite peculiar.Laplace's demon - Wikipedia, Charles Darwin (1875): The Variation of Animals and Plants Under Domestication, Andrey Nikolaevich Kolmogorov (1956): Foundations of the Theory of Probability
A traditional source of randomness in all walks of life is the toss of a coin. As we throw the coin in the air we make it spin, mapping itself onto itself an unpredictable number of times until it lands and comes to rest with one side showing. One bit of entropy has become observable through this random process. The energy to spin the coin has come from the coin tosser. Here we imagine that the initial singularity is a real substance, capable of mapping onto itself and so executing the Brouwer fixed point theorem, a formal or abstract process not yet involving energy. Because the symmetry is structureless it has no control over its action so its action is random. This random creation of abstract Hilbert space takes the place of the quantum fluctuation proposed in standard physics. From our point of view space-time and its content does come into existence until Step 6 below. Action exists at this stage, but no energy which is the product of action and time reflected in the fundamental equation of quantum mechanics, E = hf = h/t Brouwer fixed point theorem - Wikipedia, Quantum fluctuation - Wikipedia, Planck-Einstein relation - Wikipedia
This idea of abstract cognitive creation is implicit in the ancient explanation of the doctrine of the Trinity developed by the Christian theologian Augustine of Hippo in the fifth century. He understood, through introspection, that he carried an image of himself in his mind. He proposed that the image resulting from God’s reflection on themself is real rather than imaginary and therefore divine, the second person of the Trinity, the Son. The third person of the Trinity, the Spirit, arose from the Father and the Son reflecting on themselves. The doctrine of the Nicene Council promulgated as the Nicene creed limits this process to the three persons of the traditional Trinity. Here we we see no reason for a limit and propose an analogous process for the creation of the bases of a Hilbert space of any dimension in the initial singularity. Mary Sirridge (1999): "Quam videndo intus dicimus": Seeing and Saying in De Trinitate XV Aurelius Augustine (419, 1991): The Trinity, Dale Tuggy (Stanford Encyclopedia of Philosophy): History of Trinitarian Doctrines
Return to Toc2. How can we imagine a quanta of action?
We can use the ancient theology of angels recounted by the medieval Christian theologian Thomas Aquinas to imagine the quanta of action that form the bases of Hilbert space. Aquinas was working in the philosophical picture of reality developed by Plato and Aristotle now called hylomorphism, Greek for matter-formism. Plato followed Parmenides, who considered true reality to be eternal and motion to be an illusion. He imagined that the structure of the world is a poor realization of a set of invisible, perfect immutable forms. Aristotle, Plato’s student, sought to emphasize the reality of the observable world and to explain motion and change. To do this he brought Plato’s forms down to earth by connecting them with matter. Substantial change occurs when some matter changes its form, as when a bronze sword is recast into a ploughshare. Hylomorphism - Wikipedia
Many theological traditions have included angels as messengers from the gods. Aquinas follows tradition in describing angels as incorporeal beings, living immaterial entities like Plato’s forms, invisible to us, but created by God because they are a necessary component of a perfect world. Because they are not differentiated by matter, every angel is a unique species. Mathematically we say that they are orthogonal, like the orthogonal sides of a cube. Another property of vectors in Hilbert space is that their inner products with themselves are 1, they are normalized. No matter how complex an “angelic form” may be they are all the same "size". Aquinas, Summa I, 50, 1: Are angels altogether incorporeal?, Aquinas, Summa I, 50, 4: Is every angel a different species?
There is no physical space or time in Hilbert space, only the simple abstract logic of complex linear algebra. In the real space and time which we inhabit the quantum of action is a unit of angular momentum or spin. In abstract Hilbert space we may imagine each angelic vector as a complex basis for the space. Even the complex numbers themselves can be imagined as little vectors which we represent by z, the sum of a real and an imaginary number, that is z = x + iy where i is the imaginary unit, the square root of −1: i × i = −1. This is not easy to imagine, but the Hilbert world obeys a definite set of mathematical rules which serve to explain the formal structure of the visible world. Complex number - Wikipedia, Linear algebra - Wikipedia, Dot product - Wikipedia, Inner product space - Wikipedia
Every complex number z = x + iy has a complex conjugate, often written with a superscript like z*: z* = x − iy. When a complex number is added to its conjugate the complex components iy cancel out and we left with 2x. If we multiply a complex number by its complex conjugate, we are left with its “square", a real number that measures its size or magnitude. x2 + y2. Complex conjugate - Wikipedia
We can visualize complex numbers as vectors by plotting them on a complex plane similar to a Cartesian plane. We represent real numbers on the horizontal x axis and complex numbers on the vertical y axis. This representation reveals the real beauty of complex vectors and quanta of action by showing how they can represent the periodic functions we use to represent the waves of sound, music, speech and the interactions of the “angels” we have chosen as an analogy for the complex vectors in Hilbert Space. Complex plane - Wikipedia
All the basis vectors of Hilbert space are normalised to 1, a real number which may be the radius r of a circle centred at the origin of the complex plane. Each vector zrepresents a point on this circle. This point is localized on the circle by the angle θ between the positive direction of the x axis and the radius pointing to z. We can therefore write z = rθ. By varying r and θ this notation can represent any complex number.
This representation of complex numbers makes them perfect for representing waves. If we imagine a right angled triangle drawn on the complex plane with a hypotenuse of length 1 running from the origin to the point z we see that the pythagorean theorem tells us that the square of cosine(θ) (the base of the triangle on the x axis), plus the square of the isine( θ) (the vertical side of the triangle) = 1, i.e. cos(θ)2 − sin(θ)2) = 1.
This brings us to Euler’s formula, what Richard Feynman calls one of the most remarkable, almost astounding, formulas in all of mathematics:
eiθ = cos(θ) + isin((θ).
A wave evolves as the angle θ revolves around the complex plane.
Quanta of action and vectors in Hilbert space are hard to imagine but their mathematical structure shows that they are wavelike. Perhaps the best way to think of them is as sounds, waves, phonemes, elements of speech and music. Like angels there can be a countable infinity of different individuals ranging from simple pure tones to very long speeches, the structure of a human body or a planet, and ultimately the universe. Angels and forms are some of the subtlest entities ever imagined. This is why we think of Plato as a genius. His theory of forms is very similar to quantum mechanics. What we have added to Plato’s ideas, made possible by the magic of complex numbers, are the mathematical rules for manipulating forms to show how they can be connected to make new forms. The vectors in Hilbert space are like are abstract models of the concrete genes built into living DNA and RNA. Quantum mechanics and energy realize these of abstract forms as concrete observable particles as we discuss in Steps 3 and 4.
Return to Toc3. Von Neumann axioms of abstract Hilbert Space.
The Hilbert space that emerged in the initial singularity by reflecting upon itself like Augustine’s God is the mathematical foundations of quantum mechanics. This was recognized in the 1920s by Heisenberg, Born Schrödinger, Dirac and von Neumann who constructed the first useful mathematical models of the quantum behaviour of the world. Werner Heisenberg (1925): Quantum-theoretical re-interpretation of kinematic and mechanical relations, Schrödinger equation - Wikipedia, Anthony Duncan (2024_06_04): Von Neumann’s 1927 Trilogy on the Foundations of Quantum Mechanics. Annotated Translations, P. A. M. Dirac (1983): The Principles of Quantum Mechanics (4th ed), John von Neumann (2018): The Mathematical Foundations of Quantum Mechanics
John von Neumann put the story on a sound mathematical footing. He defined an abstract n dimensional dimensional Hilbert space (Cn) of complex vectors with the three following axioms. This work is treated by the area of mathematics known as linear algebra. Linear algebra - Wikipedia
The axioms are:
Further axioms are required if we wish to apply quantum mechanics in continuous spaces, but this is unnecessary as we shall see in section 4.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; and
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".
Von Neumann notes that these axioms are very similar to those used in 3D Euclidean space, with two differences; there is no limit on the number of dimensions, and the space uses complex numbers, z = x + iy
In Step 3 we will see how quantum mechanics emerges almost automatically in this beautiful and simple space, which serves as a natural foundation for music, speech and all forms of communication.
Return to Toc4. Hilbert space lies beneath the Minkowski space in which we live
Much of the difficulty of quantum mechanics arises from fact that it is invisible us. Like the invisible minds of the things, animals and people around us, we have to guess what is going on inside them by observing their visible behaviour.
Isaac Newton developed classical mechanics and was able to produce a very effective model of the solar system using astronomical data gathered over thousands of years in 3D Euclidean space. He assumed that the same time operated throughout the world. This view was rarely questioned until people began to study electrical phenomena. James Clerk Maxwell assembled all the experimental facts that had been gathered by workers such as Faraday, Ampere, Volta, Gauss and many others into a relatively simple set of differential equations known as Maxwell’s equations which suggested that light is itself an electromagnetic phenomenon. This conjecture was shown to be true by Heinrich Hertz, laying the foundation for “wireless” communication. Classical mechanics - Wikipedia, Maxwell's equations - Wikipedia
Faraday’s law of induction tells us that there is a close relationship between electric currents and moving magnetic fields. Even though the essential point is the relative motion between the magnetic field and the wire carrying the current, different explanations emphasised either the motion of the wire or the notion of the magnet. From this Einstein concluded that the laws of mechanics and electrodynamics are mutually consistent but exhibit no property that corresponds to absolute rest. From this he draws two postulates, a principle of relativity and the postulate that in empty space light moves at a constant speed, c, independent of the motion of its source. Albert Einstein (1905): On the Electrodynamics of Moving Bodies
Some remarkable properties arise from these postulates, particularly for bodies moving at significant fractions of the speed of light. The first is the numerical relationship E = mc2 between energy and mass. The second is that an observer watching a body moving past at high speed observes that a massive body becomes heavier as its speed increases, theoretically becoming infinite at the speed of light. Only inherently massless bodies, like photons, can travel at light speed.
Further an observer able to estimate the flow of time on a fast moving body (by observing, for instance, the rate of decay of radioactive substances) will see that time slows down as the body moves faster, and comes to a standstill at the speed of light. Finally, the apparent length of a material body becomes shorter as its speed increases, becoming zero at the speed of light. From this we might conclude that a photon moving at the speed of light effectively exists outside spacetime. It is moving on a path with zero spacetime length, so that its point of creation is the same as its point of annihilation.
Hermann Minkowski packed all this information into the metric of spacetime which became the starting point for Einstein’s general theory of relativity. We will explain the quantum mechanical source of these properties of objects in spacetime in subsequent pages of this commentary as we explore the connection between Hilbert space and Minkowski space. Minkowski spacetime - Wikipedia, Salart, Baas, Branciard, Gisin, & Zbinden 2008: Testing spooky action at a distance Return to Toc5. The logical completeness of Hilbert space that renders the universe divine
Although it may have an unlimited number of dimensions, Hilbert space is much simpler and more primitive than classical Minkowski space. It has no distance, momentum, time or energy. It is more like an abstract mathematicul space of sounds, speech or music, all structures built from waves, capable of speaking anything speakable, embedded in the initial singularity. Since the singularity is structureless is has no self control so its omnipotent actions are random. The Hilbert space it creates within itself will therefore be random, providing the variation necessary for creative evolution. Since initial symmetry is eternal it can explore an unlimited space of variations. Since the world exists, it must have ultimately arrived, by trying every consistent possibility, at the structure of the universe we now inhabit. John S. Bell (1987): Speakable and Unspeakable in Quantum Mechanics.
We may therefore imagine that this evolving world ultimately fills the complete logical space of consistency. From Aquinas’s point of view, this is equivalent to saying that the universe is omnipotent and therefore divine. Aquinas, Summa I, 25, 3: Is God omnipotent?
Return to TocRevised Tuesday 30 June 2026.
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Notes and references
Further reading
Books
Augustine (419, 1991), and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 399-419, 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augustine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.
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Bell (1987), John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality . . . . This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality.
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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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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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Kolmogorov (1956), Andrey Nikolaevich, and Nathan Morrison (Translator) (With an added bibliography by A T Bharucha-Reid), Foundations of the Theory of Probability, Chelsea 1956 Preface: 'The purpose of this monograph is to give an axiomatic foundation for the theory of probability. . . . This task would have been a rather hopeless one before the introduction of Lebesgue's theories of measure and integration. However, after Lebesgue's publication of his investigations, the analogies between measure of a set and mathematical expectation of a random variable became apparent. These analogies allowed of further extensions; thus, for example, various properties of independent random variables were seen to be in complete analogy with the corresponding properties of orthogonal functions . . .'
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von Neumann (2018), John, and Nicholas A. Wheeler (editor), Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 2018 ' Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics--a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. In this new edition of this classic work, mathematical physicist Nicholas Wheeler has completely reset the book in TeX, making the text and equations far easier to read. He has also corrected a handful of typographic errors, revised some sentences for clarity and readability, provided an index for the first time, and added prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson. The result brings new life to an essential work in theoretical physics and mathematics.'
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Links
, 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 |
Albert Einstein (1905), On the Electrodynamics of Moving Bodies, An english translation of the paper that founded Special relativity. 'Examples of this sort, [in the contemporary application of Maxwell's electrodynamics to moving bodies] together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.' back |
Albert Einstein (1933), Herbert Spencer Lecture 1933: On the Method of Theoretical Physics , ' 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 |
Alexis Szejnoga (2013), Persian Perspective on Prima Philosophia: The influence of Avicenns's interpetation of Aristotelian ontology in the De ente et essentia of Saint Thomas Aquinas, Chapter 1: The wider context of the De ente et essentia In this chapter, we will take a look at the context in which the De ente et essentia was written. The first part will focus on historical-philosophical aspects, treating topics like the renewed interest in Aristotelian philosophy, the attention being paid to translations of Arabian works on theology and philosophy, notably of Avicenna and Averroës, and to two philosophical treatises, the Fons vitae and the Liber de causis, which were at the time wrongly attributed to Aristotle and an unknown Christian theologian. In addition, we will look at some bibliographical details of the De ente et essentia: an approximation of the period in which it was written, the rationale behind its structure, a summary of its intent and scope, and its reception among medieval and modern philosophy.' back |
Anthony Duncan (2024_06_04), Von Neumann’s 1927 Trilogy on the Foundations of Quantum Mechanics. Annotated Translations, ' Chapter 0 Introduction In his witty and insightful biography of John von Neumann (born December 28, 1903 in Budapest, as Neumann, Janos Lajos), Norman Macrae \autocite*Macrae:1992 suggests that von Neumann had realized “nearly all his achievements while he was mainly engaged in something else.” Perhaps the best illustration of this paradox can be found in the year 1927, when von Neumann, while completing the requirements for his habilitation (basically, approval to lecture officially at a German University) at the Friedrich-Wilhelm University in Berlin by submitting two separate theses, one on mathematical logic (“The axiomatization of set theory”), the other in functional analysis (“General eigenvalue theory of symmetric functional operators”), also managed to produce three papers laying the foundations for a mathematically rigorous and conceptually coherent formulation of quantum theory. The object of the present article is to present new, fully annotated translations of these remarkable papers.1 We begin with some brief biographical information on von Neumann; this will be followed by a short discussion of the scientific context of von Neumann’s incursion into physics; and, finally, a (very condensed!) description of the contents of each paper of the trilogy. back |
Apophatic theology - Wikipedia, Apophatic theology - Wikipedia, the free encyclopedia, 'Apophatic theology (from Greek ἀπόφασις from ἀπόφημι - apophēmi, "to deny")—also known as negative theology or via negativa (Latin for "negative way")—is a theology that attempts to describe God, the Divine Good, by negation, to speak only in terms of what may not be said about the perfect goodness that is God. It stands in contrast with cataphatic theology.' back |
Aquinas, Summa I, 15, 1, Are there ideas in God?, ' . . . in other agents (the form of the thing to be made pre-exists) according to intelligible being, as in those that act by the intellect; and thus the likeness of a house pre-exists in the mind of the builder. And this may be called the idea of the house, since the builder intends to build his house like to the form conceived in his mind. As then the world was not made by chance, but by God acting by His intellect, as will appear later, there must exist in the divine mind a form to the likeness of which the world was made. And in this the notion of an idea consists.' back |
Aquinas, Summa I, 25, 3, Is God omnipotent?, '. . . God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.' back |
Aquinas, Summa I, 50, 1, Are angels altogether incorporeal?, 'I answer that, There must be some incorporeal creatures. For what is principally intended by God in creatures is good, and this consists in assimilation to God Himself. And the perfect assimilation of an effect to a cause is accomplished when the effect imitates the cause according to that whereby the cause produces the effect; as heat makes heat. Now, God produces the creature by His intellect and will (14, 8; 19, 4 ). Hence the perfection of the universe requires that there should be intellectual creatures. Now intelligence cannot be the action of a body, nor of any corporeal faculty; for every body is limited to "here" and "now." Hence the perfection of the universe requires the existence of an incorporeal creature.' 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, 3, 4, Are essence and existence the same in God?, ' I answer that, God is not only His own essence, as shown in the preceding article, but also His own existence. This may be shown in several ways. First, whatever a thing has besides its essence must be caused either by the constituent principles of that essence (like a property that necessarily accompanies the species — as the faculty of laughing is proper to a man — and is caused by the constituent principles of the species), or by some exterior agent — as heat is caused in water by fire. Therefore, if the existence of a thing differs from its essence, this existence must be caused either by some exterior agent or by its essential principles. Now it is impossible for a thing's existence to be caused by its essential constituent principles, for nothing can be the sufficient cause of its own existence, if its existence is caused. Therefore that thing, whose existence differs from its essence, must have its existence caused by another. But this cannot be true of God; because we call God the first efficient cause. Therefore it is impossible that in God His existence should differ from His essence.' back |
Brouwer fixed-point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compact convex set to itself, there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset K of Euclidean space to itself.' back |
Charles Darwin (1869), The Variation of Animals and Plants under Domestication, ' 'In scientific investigations . . . it is permitted to invent any hypothesis, and if it explains various large and independent classes of facts, it rises to the rank of a well grounded theory.' back |
Classical mechanics - Wikipedia, Classical mechanics - Wikipedia, the free encyclopedia, '' The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on the 17th century foundational works of Sir Isaac Newton, and the mathematical methods invented by Newton, Gottfried Wilhelm Leibniz, Leonhard Euler and others to describe the motion of bodies under the influence of forces. Later, methods based on energy were developed by Euler, Joseph-Louis Lagrange, William Rowan Hamilton and others, leading to the development of analytical mechanics (which includes Lagrangian mechanics and Hamiltonian mechanics). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics' back |
Complex conjugate - Wikipedia, Complex conjugate - Wikipedia, the free encyclopedia, In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a − bi. The complex conjugate of zis often denoted as z∗. 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 |
Complex plane - Wikipedia, Complex plane - Wikipedia, the free encuclopedia, ' In mathematics, the complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.' back |
Dale Tuggy (Stanford Encyclopedia of Philosophy), Trinity, ' A Trinity doctrine is commonly expressed as the statement that the one God exists as or in three equally divine “Persons”, the Father, the Son, and the Holy Spirit. Every term in this statement (God, exists, as or in, equally divine, Person) has been variously understood. The guiding principle has been the creedal declaration that the Father, Son, and Holy Spirit of the New Testament are consubstantial (i.e. the same in substance or essence, Greek: homoousios). Because this shared substance or essence is a divine one, this is understood to imply that all three named individuals are divine, and equally so. Yet the three in some sense “are” the one God of the Bible.' back |
Dot product - Wikipedia, Dot product - Wikipedia, the free encyclopedia, 'In mathematics, the dot product, or scalar product (or sometimes inner product in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and then summing those products. The name "dot product" is derived from the centered dot "." that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar (rather than vector) nature of the result.' back |
Euler's formula - Wikipedia, Euler's formula - Wikipedia, the free encyclopedia, 'Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics". (Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley, p. 22-10. ISBN 0-201-02010-6.) back |
Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' 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. |
Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, ' In the history of science, Laplace's demon was a notable published articulation of causal determinism on a scientific basis by Pierre-Simon Laplace in 1814. According to determinism, if someone (the demon) knows the precise location and momentum of every particle in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics. |
Law of noncontradiction - Wikipedia, Law of noncontradiction - Wikipedia, the free encyclopedia, ' In logic, the law of noncontradiction (LNC; also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that for any given proposition, the proposition and its negation cannot both be simultaneously true, e.g., the proposition "the house is white" and its negation "the house is not white" are mutually exclusive. To express the fact that the law is tenseless and to avoid equivocation, sometimes the law is amended to say "contradictory propositions cannot both be true 'at the same time and in the same sense'".' back |
Linear algebra - Wikipedia, Linear algebra - Wikipedia, the free encyclopedia, ' Linear algebra is the branch of mathematics concerning linear equations such as a1 x1 + ⋯ + anxn = b.
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to function spaces.' 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 |
Maxwell's equations - Wikipedia, Maxwell's equations - Wikipedia, the free encyclopedia, ' Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.' back |
Minkowski spacetime - Wikipedia, Minkowski spacetime - Wikipedia, the free encyclopedia, ' By1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 + y2 + z2 − c2 t2.' back |
Quantum fluctuation - Wikipedia, Quantum fluctuation - Wikipedia, the free encyclopedia, 'In quantum physics, a quantum fluctuation (or vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. They are tiny random fluctuations in the values of the fields which represent elementary particles, such as electric and magnetic fields which represent the electromagnetic force carried by photons, W and Z fields which carry the weak force, and gluon fields which carry the strong force. Vacuum fluctuations appear as virtual particles, which are always created in particle-antiparticle pairs.' back |
Salart, Baas, Branciard, Gisin, & Zbinden 2008, Testing spooky action at a distance, ' In science, one observes correlations and invents theoretical models that describe them. In all sciences, besides quantum physics, all correlations are described by either of two mechanisms. Either a first event influences a second one by sending some information encoded in bosons or molecules or other physical carriers, depending on the particular science. Or the correlated events have some common causes in their common past. Interestingly, quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This new kind of cause reveals itself, e.g., in correlations that violate Bell inequalities (hence cannot be described by common causes) between space-like separated events (hence cannot be described by classical communication). Einstein branded it as spooky action at a distance. A real spooky action at a distance would require a faster than light influence defined in some hypothetical universally privileged reference frame. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test during more than 24 hours between two villages separated by 18 km and approximately east-west oriented, with the source located precisely in the middle. We continuously observed 2-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of this spooky influence. For instance, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10^-3 that of the speed of light, then the speed of this spooky influence would have to exceed that of light by at least 4 orders of magnitude. back |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, ' In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . . In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Thomas Aquinas, De Ente et Essentia, Latin text of one of Thomas Aquinas early works. |
Werner Heisenberg (1925), Quantum-theoretical re-interpretation of kinematic and mechanical relations , 'The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.' [From Sources of Quantum Mechanics, edited by B. L. van der Waerden (Amsterdam, North-Holland, 1967)] back |