page 6: Symmetry
Universals
One of the most remarkable features of the world is that we are able to learn and communicate 'laws of nature' which help us survive. Our knowledge of gravitation is built into us as soon as we begin to move ourselves about, helped by the pain of numerous falls. One of the feature of knowledge is that one item of knowledge can apply to many instances. In this way knowledge acts as an algorithm, a fixed mathematical structure (like multiplication) which applies in an unlimited number of particular cases. Algorithm - Wikipedia
This feature of knowledge, that one idea can apply to many instances, gave rise to the 'problem of universals', which has been debated by philosophers since antiquity. Plato held that universals, like humanity, roundness and so on, exist as real and perfect forms which we cannot see, but which are responsible for the similarities in our world. Aristotle brought Plato's forms down to Earth. Moving in the opposite direction, 'nominalists' maintain that universals are merely a human mental construct and do not exist in reality. The world is concrete, they say, and every individual is unique. Problem of universals - Wikipedia, Richard Kraut - Plato, Christopher Shields - Aristotle, Nominalism - Wikipedia
Here we embrace both views: there are symmetries in the world which underlie our knowledge, but all observable symmetries are broken, which accounts for the concrete nature of the world.
'Laws of nature'
Traditionally, science is the search for the laws of nature, things that stay the same while all else changes around them. Our existence is made possible by such laws. All our engineering and technology depend upon them. When we build a bridge we want it to be a fixed point, a law in effect, standing up and doing its job under all circumstances. The stability of the bridge embodies many other laws about the nature of chemical bonding and the distribution of stresses and strains in structural elements. A bridge, like all technology, is an application of the laws of nature as revealed by science and experience. Physical law - Wikipedia
The term 'law of nature' has been formed by analogy to human laws developed by customs, parliaments or monarchs. A law is a fixed point, often recorded in a written document accessible to everyone, which requires or prohibits various activities. You must pay tax. You must not kill. The political theory of the rule of law maintains that we are all equal before the law. A law is a symmetry of a society. It is (in principle) the same for everybody. Walker: The Rule of Law, Rule of Law - Wikipedia
Over the last hundred years or so the term 'law' in the physical sciences has gradually been replaced with the words like symmetry, invariance and conservation. These terms point to things that do not change with time and can therefore be represented by fixed mathematical structures and be relied upon in daily life. The existence of invariants in the Universe answers the problem raised by Parmenides about the possibility of certain knowledge in a changing world. Symmetry - Wikipedia, Invariant (physics) - Wikipedia, Conservation law - Wikipedia. Parmenides - Wikipedia
Symmetry and continuity
Symmetry and continuity are closely related. They are two different ways of saying nothing visible happens even though we might imagine some invisible change such as the rotation of a featureless geometrically perfect wheel. We can only tell that such a wheel is turning by placing a mark upon it, thus breaking its continuity or symmetry.
The formal connection between symmetry, invariance and conservation is was established by Emmy Noether. Neuenschwander sees Noether's theorem as the central organizing principle of advanced physics. (page xi). Emmy Noether: Invariant variation problems, Noether's theorem - Wikipedia, Dwight E. Neuenschwander: Emmy Noether's Wonderful Theorem
Noether's theorem states that every differentiable (that is continuous) symmetry of the action of a physical system has a corresponding conservation law. An action is said to have a symmetry with respect to a quantity if that quantity does not appear in the Lagrangian, and so makes no contribution to the action. It is ignored by the process whose action is being computed, and is therefore conserved. Lagrangian - Wikipedia
Hamilton's principle: when functionals are invariant
The first step toward Noether's theorem is Hamilton's principle. Newton worked out his laws of motion by studying the behaviour of moving bodies including planets, satellites and pendulums.
In the Newtonian formulation, a certain force on a body is considered to produce a definite motion: that is a definite effect is always associated with a certain cause. According to Hamilton's Principle, however, the motion of a body may be considered to result from an attempt by Nature to achieve a certain purpose namely to minimize the time integral of the difference between kinetic and potential energies.' Marion, quoted in Neuenschwander page 25.
Hamilton's approach derives from Maupertuis' principle, that nature acts with maximum efficiency, ie it uses the minimum action necessary to get something done. Maupertuis' principle - Wikipedia, Yourgrau & Mandelstam: Variational principles
More generally, we understand Hamilton's principle to mean that the actual path taken by a dynamic process is the one where small deviations from the path do not change the action S, that is S is stationary. Points of stationary action are discovered using the calculus of variations. The calculus of variations leads to the Euler-Lagrange equation: 'a differential equation whose solutions are the functions for which a given functional is stationary'. Action (physics) - Wikipedia, Calculus of variations - Wikipedia, Euler-Lagrange equation - Wikipedia
Noether writes:
'What is to follow . . . represents a combination of the methods of the formal calculus of variations with those of group theory.'
'By a "group of transformations", familiarly, is meant a system of transformations such that for each transformation, there exists an inverse contained in the system, and such that the composition of any two transformations of the system in turn belongs to the system.' Emmy Noether: Invariant variation problems
'Consider transformations described by a continuous parameter that can be varied starting from ε = 0. The identity transformation ε = 0 makes no change whatever. As ε increases, the differences between the original and the new coordinate system becomes larger and larger. Noether's theorem deals only with continuous transformations.' (Neuenschwander page 61) Lie Group - Wikipedia
Noether's proof depends on the continuity and differentiability of Lie groups. 'If the functional is extremal, then the Euler-Lagrange equation holds. When the functional is both invariant and extremal, then the Rund-Trautman identity together with the Euler-Lagrange equation gives immediately the Noether conservation law. In the case of our field φ(t, x), the Noether conservation law offers an equation of continuity
∂μ jμ = 0
where the current jμ stands for [a function of the] generators of a transformation .' (Neuenschwander page 107)
Conservation of action, energy and momentum
Physics recognises three fundamental and exact conservation laws, the conservation of action, angular momentum or spin; the conservation of energy; and the conservation of linear momentum. All three follow from Noether's theorem. Invariance of a functional under spatial rotation yields conservation of angular momentum; invariance under time translation yields conservation of energy; invariance under space translation yields conservation of linear momentum. Angular momentum - Wikipedia, Energy - Wikipedia, Momentum - Wikipedia
Action
Familiarly, we understand action to be the fundamental observable in the universe. It is common to all events that they are an action, that is something happens or is done. This notion harks back to the philosophers of antiquity. Aristotle understood the first unmoved mover to be pure action. This idea became the medieval Christian God of pure action documented by Thomas. Here we apply the same idea, that the universe is action, a form of perpetual motion. Aquinas 13: Does God Exist?
Quantum mechanics has introduced the quantum of action, measured by Planck's constant, which is believed to have a definite and unchanging value. Every quantum event involves an integral number of quanta of action. In the emission or reception of a photon by an atomic electron, we see that the electron loses or gains one quantum of orbital angular momentum and emits or absorbs a photon with one quantum of spin. Spontaneous emission - Wikipedia
Newton established that action and reaction are equal and opposite. On this site we propose a network model and see every event as an act of communication. We understand that events occur when the space-time interval between their participants is zero. The act of communication affects both participants, as we see in the quantum theory of measurement. Quanta of action are conserved in effect by Newton's third law. Measurement in quantum mechanics - Wikipedia
Since we measure quanta of action with quanta of action, we can attribute no particular size to a quantum of action, but find that they are all equal when measured in terms of the macroscopic dimensions Mass, Length and Time, so we write h = ML2T-1. All quanta of action are symmetrical with respect to size. Dimensional analysis - Wikipedia
Energy
We may think of a 'bare' action as simply an atomic event without any particular nature. Such events can be counted. Here we accept Landauer's idea that information is physical. The information carried by a quantum of action may be modelled simply as arithmetic units. The normal physical measure of dimensionless quantities like this is entropy. Normally we express entropy as the logarithm of a count of states, echoing Bolzmann's equation S = k log W. Boltzmann's entropy formula - Wikipedia
Energy is closely related to time by the Planck-Einstein relationship E = hf where f, frequency, is the inverse of time and h is Planck's constant. In other words, energy is simply the time rate of action.
The simplest way to think of a communication network is as a closed system of leakproof pipes. From a physical point of view, the fluid flowing in the pipes may be action, energy or momentum. Conservation of these quantities is guaranteed because the pipes are leakproof. From this point of view, the energy of a link is equal to the rate of flow of action in the link. We may think of each exchange of a message as an action so that the bandwidth of a channel is equal to its energy.
Momentum
As Newton noted, a body at rest remains at rest and a body in motion continues to move in a straight line unless acted upon by a force. This statement captures the nature of linear momentum. Linear momentum is the rate of change of action with distance, expressed by the de Broglie equation p = h/λ where λ is wavelength.
Symmetry in the quantum world
Noether's theorem and the physics of symmetry and conservation are intimately linked to the continuity of Lie groups. The paradigmatic representation of the continuum is the real numbers which can be placed into one to one correspondence with all the points in the real line. Real line - Wikipedia
Although most scientists seem to have thought since time immemorial that the universe is continuous, all the evidence says otherwise. Quantum mechanics tells us that there is an indivisible atomic action, and common experience tells us that all the objects and events in our world are discrete, having some sort of fixed size and duration. The only evidence for continuity is that macroscopic motion appears to be continuous and continuous mathematics has turned out to be very useful for modelling the universe in quantum mechanics and relativity where calculus is involved. But it may not be the last word. Here we are exploring the idea that the universe is digital to the core.
To see how this might work, we introduce the notion of logical continuity. The paradigm of a logical continuum is deterministic computer executing a computable function. Such a machine establishes a logical connection between an input state, the starting position of the computer, and a final state, when the computation is complete. Turing machine - Wikipedia
An instance of a logical continuum is Weierstrasse's definition of a continuous function. The geometric line serves as a starting point for the study of continuity. It was discovered long ago that there are geometrical line segments (like the diagonal of a unit square) which cannot be measured with natural or rational numbers. This observation led to the development of the real numbers and Weierstrasses' epsilon-delta definition of a continuous function which formalizes the idea that small changes in the independent variable of a continuous function should cause only small changes in the dependent variable. This is very similar to the ancient idea that a continuum is infinitely divisible, it contains no atoms, no jumps. Epsilon and delta can smoothly approach zero as closely as we like. Nevertheless, both ε and δ in the definition of continuity remain definite and separate quantities. Further examination suggests that all the mathematical arguments about continuity, such as those which underlie Noether's theorem, are logical continua. This suggests that logical continuity is prior to and deeper than geometric continuity. It is interesting that the epsilon-delta argument for continuity has a circular feel since it depends upon the continuity of the quantities epsilon and delta themselves! Continuous function - Wikipedia
The link between the continuous mathematics of the Lie group used by Noether and the digital transfinite computer network lies in the role played by groups and reversibility in the coding and decoding of messages. In communication terms, the combination of the operation O and its inverse, O-1 which yields the identity corresponds to a 'coder-decoder' or 'codec'. From a communication point of view, this is the group expression of 'nothing happens'. The message arrives as it was sent, unchanged. Because group operations are reversible, entropy is conserved, that is the order (cardinal) of a group is not changed by the group operations. Group (mathematics) - Wikipedia, Codec - Wikipedia, Richard Feynman: Symmetry and conservation,
This suggests that we think of a symmetry as an algorithm, something that does not change even though its inputs and outputs may change. The arithmetic algorithm for multiplication, for instance, applies to all numbers, no matter how large or small.
From this point of view, the layered network is a hierarchy of symmetries, the simple algorithms at the bottom underlie the more complex algorithms in higher layers. We can see this structure in in Turing's development of the computing machine, where he treated simple algorithms as subroutines in larger algorithms and used this system to build a computer that is generally agreed to be able to compute anything computable. It has become the definition of computability. Alan Turing
Symmetry breaking: the boundary between past and future
The mathematical foundation of quantum mechanics, worked out in mathematical complex function theory, is invisible. It cannot be observed. It is also considered to be deterministic and reversible, so that at least as far as the mathematics goes, Laplace's Demon could predict the whole future. Because it is reversible, entropy is conserved and so we can say that the mathematical model of quantum mechanics is not creative. Laplace's demon - Wikipedia
Quantum mechanics is a group theory, up until the observation, at which point the group symmetry is broken, and the system takes an irreversible step. The symmetry is broken and a boundary is established between past and future. Here we have a explanation for the 'arrow of time' in terms of the symmetry of quantum mechanics. The collapse of the wave function comes after the establishment of the wave function. The symmetry of past and future is broken. Wave function collapse - Wikipedia
Creation
The ancient doctrine of the Trinity explains how the single divine personality of the Hebrew Bible becomes the trinity of persons in the Christian New Testament. This transition was possibly the greatest problem faced by Christian theologians. Their approach to its solution was psychological.
It seems obvious enough that to know something is to contain it in some way. We often express the things that we know in words, and so it was natural enough to consider the source of our word to be an inner word, idea, concept or memory. Augustine and Aquinas, reading the first verse of John's Gospel, came to the conclusion that God the Father's idea of himself was identically God, but a distinct personality, we might say God, but not this God. The Father and the Son share the divine symmetry, but it is broken to give two separate persons. John 1:1 - Wikipedia
Here we see the initial singularity predicted by the general theory of relativity as formally identical to the classical model God developed by Thomas Aquinas. We assume that the initial singularity, like the classical God, can be understood as pure action. We assume that, like the classical God, this initial symmetry is eternal, that is contains no time. Then like the origin of the Trinity, this symmetry is broken to create time and energy and described in page 4: Energy
(revised 20 February 2020)
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Further readingBooks
Neuenschwander, Dwight E, Emmy Noether's Wonderful Theorem, Johns Hopkins University Press 2011 Jacket: A beautiful piece of mathematics, Noether's therem touches on every aspect of physics. Emmy Noether proved her theorem in 1915 and published it in 1918. This profound concept demonstrates the connection between conservation laws and symmetries. For instance, the theorem shows that a system invariant under translations of time, space or rotation will obey the laws of conservation of energy, linear momentum or angular momentum respectively. This exciting result offers a rich unifying principle for all of physics.'
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Walker, Geoffrey de Q, The Rule of Law: Foundations of Constitutional Democracy, Melbourne University Press 1988 Jacket: 'The author argues that the survival of any useful rule of law model is currently threatened by distortions in the adjudication process, by perversion of law enforcement (by fabrication of evidence and other means), by the excessive production of new legislation with its degrading effect on long-term legal certainty and on long-standing safeguards, and by legal theories that are hostile to the very concept of rule of law. In practice these trends have produced a great number of legal failures from which we must learn.'
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Yourgrau, Wolfgang, and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Variational principles serve as filters for parititioning the set of dynamic possibilities of a system into a high probability and a low probability set. The method derives from De Maupertuis (1698-1759) who formulated the principle of least action, which states that physical laws include a rule of economy, the principle of least action. This principle states that in a mathematically described dynamic system will move so as to minimise action. Yourgrau and andelstam explains the application of this principle to a variety of physical systems.
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Links
Action (physics) - Wikipedia, Action (physics) - Wikipedia, the free encyclopedia, 'In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has the dimensions of energy.time or momentum.length], and its SI unit is joule-second.' back |
Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by some finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable of a real or computable variable, computable predicates and so forth. . . . ' back |
Algorithm - Wikipedia, Algorithm - Wikipedia, the free encyclopedia, ' As an effective method, an algorithm can be expressed within a finite amount of space and time,[3] and in a well-defined formal language[4] for calculating a function.[5] Starting from an initial state and initial input (perhaps empty),[6] the instructions describe a computation that, when executed, proceeds through a finite[7] number of well-defined successive states, eventually producing "output"[8] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.' back |
Aquinas 13, 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. . . . The second way is from the nature of the efficient cause. . . . The third way is taken from possibility and necessity . . . The fourth way is taken from the gradation to be found in things. . . . The fifth way is taken from the governance of the world. back |
Boltzmann's entropy formula - Wikipedia, Boltzmann's entropy formula - Wikipedia, the free encyclopedia, 'In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate:
S = k ln W
where k is the Boltzmann constant, . . . which is equal to 1.38062 x 10−23 J/K. back |
Calculus of variations - Wikipedia, Calculus of variations - Wikipedia, the free encylopedia, 'Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.' back |
Christopher Shields, Aristotle (Stanford Encyclopedia of Philosophy), First published Thu Sep 25, 2008
'Aristotle (384–322 B.C.E.) numbers among the greatest philosophers of all time. Judged solely in terms of his philosophical influence, only Plato is his peer: . . . A prodigious researcher and writer, Aristotle left a great body of work, perhaps numbering as many as two-hundred treatises, from which approximately thirty-one survive. His extant writings span a wide range of disciplines, from logic, metaphysics and philosophy of mind, through ethics, political theory, aesthetics and rhetoric, and into such primarily non-philosophical fields as empirical biology, where he excelled at detailed plant and animal observation and taxonomy. In all these areas, Aristotle's theories have provided illumination, met with resistance, sparked debate, and generally stimulated the sustained interest of an abiding readership.' back |
Christopher Shields - Aristotle, Aristotle (Stanford Encyclopedia of Philosophy), First published Thu Sep 25, 2008
'Aristotle (384–322 B.C.E.) numbers among the greatest philosophers of all time. Judged solely in terms of his philosophical influence, only Plato is his peer: . . . A prodigious researcher and writer, Aristotle left a great body of work, perhaps numbering as many as two-hundred treatises, from which approximately thirty-one survive. His extant writings span a wide range of disciplines, from logic, metaphysics and philosophy of mind, through ethics, political theory, aesthetics and rhetoric, and into such primarily non-philosophical fields as empirical biology, where he excelled at detailed plant and animal observation and taxonomy. In all these areas, Aristotle's theories have provided illumination, met with resistance, sparked debate, and generally stimulated the sustained interest of an abiding readership.' back |
Codec - Wikipedia, Codec - Wikipedia, the free encyclopedia, 'A codec is a device or computer program capable of encoding or decoding a digital data stream or signal. Codec is a portmanteau of coder-decoder or, less commonly, compressor-decompressor.' back |
Conservation law - Wikipedia, Conservation law - Wikipedia, the free encyclopedia, 'In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. . . .
A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. . . .
From Noether's theorem, each conservation law is associated with a symmetry in the underlying physics. back |
Continuous function - Wikipedia, Continuous function - Wikipedia, the free encyclopedia, 'IIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called "bicontinuous".' back |
Dimensional analysis - Wikipedia, Dimensional analysis - Wikipedia, the free encyclopedia, 'In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) . . .
Any physically meaningful equation (and any inequality and inequation) must have the same dimensions on the left and right sides. Checking this is a common application of performing dimensional analysis. Dimensional analysis is also routinely used as a check on the plausibility of derived equations and computations. It is generally used to categorize types of physical quantities and units based on their relationship to or dependence on other units.' back |
Emmy Noether, Invariant Variation Probems (English Translation), M. A. Tavel’s English translation of “Invariante Variation
sprobleme,”
Nachr. d. K ̈onig. Gesellsch.
d. Wiss. zu G ̈ottingen, Math-phys. Klasse
, 235–257 (1918), which originally appeared in
Transport
Theory and Statistical Physics,1
(3), 183–207 (1971).
'The problems in variation here concerned are such as to admit
a continuous group (in Lie’s
sense); the conclusions that emerge from the corresponding
differential equations find their most
general expression in the theorems formulated in Section 1 a
nd proved in following sections.
Concerning these differential equations that arise from pro
blems of variation, far more precise
statements can be made than about arbitrary differential equ
ations admitting of a group, which
are the subject of Lie’s researches. What is to follow, there
fore, represents a combination of
the methods of the formal calculus of variations with those o
f Lie’s group theory.' back |
Energy - Wikipedia, Energy - Wikipedia, the free encyclopediax, 'In physics and other sciences, energy ,. . . is a scalar physical quantity that is a property of objects and systems which is conserved by nature. Energy is often defined as the capacity to do work. Several different forms of energy, such as kinetic, potential, thermal, electromagnetic, chemical, nuclear, and mass have been defined to explain all known natural phenomena.
Energy is converted from one form to another, but it is never created or destroyed. This principle, the conservation of energy, was first postulated in the early 19th century, and applies to any isolated system. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.' back |
Euler-Lagrange equation - Wikipedia, Euler-Lagrange equation - Wikipedia, the free encyclopedia, 'In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation although the latter name is ambiguous (see disambiguation page), is a differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph Louis Lagrange in the 1750s' back |
Group (mathematics) - Wikipedia, Group (mathematics) - Wikipedia, the free encyclopedia, 'In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.' back |
Invariant (physics) - Wikipedia, Invariant (physics)- Wikipedia, the free encyclopedia, 'In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.' back |
John 1:1 - Wikipedia, John 1:1 - Wikipedia, the free encyclopedia, 'The Greek word λόγος or logos is a word with various meanings. It is often translated into English as "Word" but can also mean thought, speech, account, meaning, reason, proportion, principle, standard, or logic, among other things. It has varied use in the fields of philosophy, analytical psychology, rhetoric and religion.' back |
Lagrangian - Wikipedia, Lagrangian - Wikipedia, the free encyclopedia, 'The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics.
In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. In symbols, L = T - V. ' back |
Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, 'We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.' A Philosophical Essay on Probabilities, Essai philosophique dur les probabilites introduction to the second edition of Theorie analytique des probabilites based on a lecture given in 1794. back |
Maupertuis' principle - Wikipedia, Maupertuis' principle - Wikipedia, the free encyclopedia, 'In classical mechanics, Maupertuis' principle (named after Pierre Louis Maupertuis) is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally stated principle of least action. More precisely, it is a formulation of the equations of motion for a physical system not as differential equations, but as an integral equation, using the calculus of variations.' back |
Measurement in quantum mechanics - Wikipedia, Measurement in quantum mechanics - Wikipedia, the free encyclopedia, 'The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus.' back |
Momentum - Wikipedia, Momentum - Wikipedia, the free encyclopedia, 'In classical mechanics, momentum (pl. momenta; SI unit kg·m/s, or, equivalently, N·s) is the product of the mass and velocity of an object (p=mv). For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.' back |
Noether's theorem - Wikipedia, Noether's theorem - Wikipedia, the free encyclopedia, 'Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.' back |
Nominalism - Wikipedia, Nominalism - Wikipedia, the free encyclopedia, 'Nominalism is a metaphysical view in philosophy according to which general or abstract terms and predicates exist, while universals or abstract objects, which are sometimes thought to correspond to these terms, do not exist. There are at least two main versions of nominalism. One version denies the existence of universals – things that can be instantiated or exemplified by many particular things (e.g., strength, humanity). The other version specifically denies the existence of abstract objects – objects that do not exist in space and time.' back |
Parmenides - Wikipedia, Parmenides - Wikipedia, the free encyclopedia, 'Parmenides of Elea (early 5th century BC) was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy, his only known work is a poem which has survived only in fragmentary form. In it, Parmenides describes two views of reality. In the Way of Truth, he explained how reality is one; change is impossible; and existence is timeless, uniform, and unchanging. In the Way of Opinion, he explained the world of appearances, which is false and deceitful. These thoughts strongly influenced Plato, and through him, the whole of western philosophy.' back |
Physical law - Wikipedia, Physical law - Wikipedia, the free encyclopedia, 'A physical law or scientific law "is a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present."' back |
Problem of universals - Wikipedia, Problem of universals - Wikipedia, the free encyclopedia, 'The problem of universals is an ancient problem in metaphysics about whether universals exist. Universals are general or abstract qualities, characteristics, properties, kinds or relations, such as being male/female, solid/liquid/gas or a certain colour[1], that can be predicated of individuals or particulars or that individuals or particulars can be regarded as sharing or participating in. . . . The problem of universals is about their status; as to whether universals exist independently of the individuals of whom they can be predicated or if they are merely convenient ways of talking about and finding similarity among particular things that are radically different. This has led philosophers to raise questions like, if they exist, do they exist in the individuals or only in people's minds or in some separate metaphysical domain?' back |
Psycholinguistics - Wikipedia, Psycholinguistics - Wikipedia, the free encyclopedia, 'Psycholinguistics or psychology of language is the study of the psychological and neurobiological factors that enable humans to acquire, use, comprehend and produce language. Initial forays into psycholinguistics were largely philosophical or educational schools of thought, due mainly to their location in departments other than applied sciences (e.g., cohesive data on how the human brain functioned). Modern research makes use of biology, neuroscience, cognitive science, linguistics, and information science to study how the brain processes language, and less so the known processes of social sciences, human development, communication theories and infant development, among others.' back |
Real line - Wikipedia, Real line - Wikipedia, the free encyclopedia, 'In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.' back |
Real number - Wikipedia, Real number - Wikipedia, the free encyclopedia, 'In mathematics, a real number is a value that represents a quantity along a continuous line. . . . The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field (R ; + ; · ; <), up to an isomorphism,' back |
Richard Feynman, Lectures on Physics III:17 Symmetry and Conservation Laws, 'The most beautiful thing of quantum mechanics is that the conservation theorems can, in a sense, be derived from something else, whereas in classical mechanics they are practically the starting points of the laws. . . . In quantum mechanics, however, the conservation laws are very deeply related to the principle of superposition of amplitudes, and to the symmetry of physical systems under various changes. This is the subject of the present chapter. Although we will apply these ideas mostly to the conservation of angular momentum, the essential point is that the theorems about the conservation of all kinds of quantities are—in the quantum mechanics—related to the symmetries of the system.' back |
Richard Kraut - Plato, Plato (Stanford Encyclopedia of Philosophy), First published Sat Mar 20, 2004; substantive revision Thu Sep 17, 2009
'Plato (429–347 B.C.E.) is, by any reckoning, one of the most dazzling writers in the Western literary tradition and one of the most penetrating, wide-ranging, and influential authors in the history of philosophy. . . . Few other authors in the history of philosophy approximate him in depth and range: perhaps only Aristotle (who studied with him), Aquinas, and Kant would be generally agreed to be of the same rank.' back |
Rule of Law - Wikipedia, Rule of Law - Wikipedia, the free encyclopedia, 'The rule of law is the legal principle that law should govern a nation, as opposed to being governed by arbitrary decisions of individual government officials. It primarily refers to the influence and authority of law within society, particularly as a constraint upon behavior, including behavior of government officials.[2] The phrase can be traced back to 16th century England, and it was popularized in the 19th century by British jurist A. V. Dicey. The concept was familiar to ancient philosophers such as Aristotle, who wrote "Law should govern".' back |
Spontaneous emission - Wikipedia, Spontaneous emission - Wikipedia, the free encyclopedia, 'Spontaneous emission is the process by which a quantum system such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy (e.g., the ground state) and emits quanta of energy.' back |
Symmetry - Wikipedia, Symmetry - Wikipedia, the free encyclopedia, 'Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to a transformation, such as reflection but including other transforms too. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.' back |
Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer.
The "machine" was invented in 1936 by Alan Turingwho called it an "a-machine" (automatic machine). The Turing machine is not intended as practical computing technology, but rather as a hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.' back |
Wave function collapse - Wikipedia, Wave function collapse - Wikipedia, the free encyclopedia, 'In quantum mechanics, wave function collapse is said to occur when a wave function—initially in a superposition of several eigenstates—appears to reduce to a single eigenstate (by "observation"). It is the essence of measurement in quantum mechanics and connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is continuous evolution via the Schrödinger equation.' back |
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