Celestial mechanics

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https://en.wikipedia.org/wiki/Celestial_mechanics

Celestial mechanics is the branch of astronomy that deals with the motions and gravitational interactions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data. The computation of the motion of the bodies through orbital mechanics can be simplified by using an appropriate inertial frame of reference. This leads to the use of various different coordinate systems, such as the Heliocentric (Sun-centered) coordinate system.

In a binary system of objects interacting through gravity, Newtonian mechanics can used to produce a set of orbital elements that will predict with reasonable accuracy the future position of the two bodies. This method demonstrates the correctness of Kepler's laws of planetary motion. Where one of the bodies is sufficiently massive, general relativity must be included to predict apsidal precession. The problem becomes more complicated when another body is added, creating a three-body problem that can not be solved exactly. Perturbation theory is used to find an approximate solution to this problem.

History

For early theories of the causes of planetary motion, see Dynamics of the celestial spheres.

Modern analytic celestial mechanics started with Isaac Newton's Principia (1687). The name celestial mechanics is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term celestial mechanics. Prior to Kepler, there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of the physical causes of the planets' motion.

Laws of planetary motion

For detailed treatments of how his laws of planetary motion can be used, see Kepler's laws of planetary motion and Kepler problem.

Johannes Kepler was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy in the 2nd century to Copernicus, with physical concepts to produce a New Astronomy, Based upon Causes, or Celestial Physics in 1609. His work led to the laws of planetary orbits, which he developed using his physical principles and the planetary observations made by Tycho Brahe. Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before Newton developed his law of gravitation in 1686.

Newtonian mechanics and universal gravitation

Isaac Newton is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like cannon balls and falling apples, could be described by the same set of physical laws. In this sense he unified celestial and terrestrial dynamics. Using his law of gravity, Newton confirmed Kepler's laws for elliptical orbits by deriving them from the gravitational two-body problem, which Newton included in his epochal Philosophiæ Naturalis Principia Mathematica in 1687.

Three-body problem

Main article: Three-body problem

The five Lagrange points for a circular orbit

Following Newton, mathematicians attempted to solve the more complex problem of predicting the future motion of three bodies interacting through gravity: the three-body problem. The first to provide a periodic solution was the Swiss mathematician Leonhard Euler, who in 1762 demonstrated three equilibrium points lie along a straight line passing through the two primary masses. If a body of infinitesimal mass occupied one of these points, it would remain there in a stable orbit. French mathematician Joseph-Louis Lagrange attempted to solve this restricted three-body problem in 1772, and discovered two more stable orbits at the vertices of equilateral triangles with the two primary masses. Collectively, these solutions became known as the Lagrange points.

Lagrange reformulated the principles of classical mechanics, emphasizing energy more than force, and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comets and such (parabolic and hyperbolic orbits are conic section extensions of Kepler's elliptical orbits). More recently, it has also become useful to calculate spacecraft trajectories.

Henri Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton's Principia.

These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).

Standardisation of astronomical tables

Simon Newcomb was a Canadian-American astronomer who revised Peter Andreas Hansen's table of lunar positions. In 1877, assisted by George William Hill, he recalculated all the major astronomical constants. After 1884 he conceived, with A. M. W. Downing, a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in Paris, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.

Anomalous precession of Mercury

Main article: Apsidal precession

Apsidal precession of Mercury's orbit around the Sun (not to scale)

In 1849, Urbain Le Verrier reported that Mercury's closest approach the Sun, its perihelion, was observed to advance at the rate of 43″ per century. This precession of Mercury's perihelion could not be accounted for by known gravitational perturbations using Newton's law. Instead, Le Verrier later attributed the effect to a proposed planet orbiting inside the orbit of Mercury. Dubbed Vulcan, subsequent searches failed to locate any such body. The cause remained a mystery until Albert Einstein explained the apsidal precession in his 1916 paper The Foundation of the General Theory of Relativity. General relativity led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in proximity to massive bodies. This led to attempts to solve the two-body problem in general relativity and the discovery of gravitational radiation.

Examples of problems

Celestial motion, without additional forces such as drag forces or the thrust of a rocket, is governed by the reciprocal gravitational acceleration between masses. A generalization is the n-body problem, where a number n of masses are mutually interacting via the gravitational force. Although analytically not integrable in the general case, the integration can be well approximated numerically.

Examples:

• 4-body problem: spaceflight to Mars (for parts of the flight the influence of one or two bodies is very small, so that there we have a 2- or 3-body problem; see also the patched conic approximation)
• 3-body problem:
  • Quasi-satellite
  • Spaceflight to, and stay at a Lagrangian point

In the ${\displaystyle n=2}$ case (two-body problem) the configuration is much simpler than for ${\displaystyle n>2}$. In this case, the system is fully integrable and exact solutions can be found.

Examples:

• A binary star, e.g., Alpha Centauri (approx. the same mass)
• A binary asteroid, e.g., 90 Antiope (approx. the same mass)

A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the orbiting body, is much smaller than the other, the central body. This is also often approximately valid.

Examples:

• The Solar System orbiting the center of the Milky Way
• A planet orbiting the Sun
• A moon orbiting a planet
• A spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)

Perturbation theory

Main article: Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to the "guess, check, and adjust" method used in numerical analysis, which is ancient.) The earliest use of modern perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun. Additional sources of orbital perturbation include atmospheric drag, solar radiation pressure, and non-uniform gravitational fields.

Perturbation methods start with a simplified form of the original problem, which is chosen to be exactly solvable. In celestial mechanics, this is usually a Keplerian ellipse, which is correct when there are only two gravitating bodies, but is often close enough for practical use. The solved, but simplified problem is then "perturbed" to make its time-rate-of-change equations for the object's position closer to the values from the real problem. The changes that result from the terms in the equations are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a better approximation.

A partially corrected solution can be re-used as the new starting point for another cycle of perturbations and corrections. In principle, the recycling of prior solutions to obtain a better solution could continue indefinitely. The difficulty is that the corrections usually progressively make the new solutions more complicated. Newton is reported to have said, regarding the problem of the Moon's orbit "It causeth my head to ache.".

Reference frame

Main article: Frame of reference

An inertial frame of reference (brown) mapped from a baseline coordinate system (blue)

A reference frame is an arbitrary defined coordinate system, whose origin, orientation, and scale are specified in physical space. The frame is aligned via a set of reference points, such as distant galaxies. Problems in celestial mechanics are often posed in simplifying reference frames, such as the synodic reference frame applied to the three-body problem, where the origin coincides with the barycenter of the two larger celestial bodies. Other reference frames for n-body simulations include those that place the origin to follow the center of mass of a body, such as the heliocentric and the geocentric reference frames. The choice of reference frame gives rise to phenomena such as the retrograde motion of superior planets in a geocentric reference frame.

An Inertial frame of reference is employed for bodies with mass. Thus a Lunar Reference System defines an Earth Inertial frame with Earth as the origin, the Lunar Inertial frame having an origin of the Moon, and an Earth-Moon Barycentric Rotating frame anchored to the rotating Earth-Moon barycenter. Positioning systems such as GPS or GLONASS use a reference frame based on the Earth. However, these are unsuitable for navigation in space. For interplanetary trajectories, a heliocentric (Sun-centered) coordinate system is used, with the XY plane aligned with the ecliptic as defined for a particular epoch.

The local standard of rest (LSR) is a reference frame based on the mean motion of stellar objects in the neighborhood of the Sun. The peculiar velocity of the Sun relative to this framework is 13.4 km/s in the direction of the solar apex. There are two possible definitions for the LSR: the first is based on the kinetic motion of nearby stars, and the second is a dynamical standard that follows the Sun in its orbit around the galaxy. These two drift apart with the passage of time as the stars follow the gravitational potential around the galaxy. Perturbations in a star's galactic orbit result in epicycle motions.

The cosmic microwave background has its own co-moving frame of reference. The relative motion of the Sun at 365 km/s results in a dipole anisotropy due to redshift of the isotropic radiation from this source.

Orbital mechanics

A satellite orbiting Earth has a tangential velocity and an inward acceleration.

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Astrodynamics is a core discipline within space-mission design and control.

Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbital plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers.

General relativity is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).

Unconscious mind

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Unconscious_mind 

In psychoanalysis and other psychological theories, the unconscious mind (or the unconscious) is the part of the psyche that is not available to introspection. Although these processes exist beneath the surface of conscious awareness, they are thought to exert an effect on conscious thought processes and behavior. The term was coined by the 18th-century German Romantic philosopher Friedrich Schelling and later introduced into English by the poet and essayist Samuel Taylor Coleridge.

The emergence of the concept of the unconscious in psychology and general culture was mainly due to the work of Austrian neurologist and psychoanalyst Sigmund Freud. In psychoanalytic theory, the unconscious mind consists of ideas and drives that have been subject to the mechanism of repression: anxiety-producing impulses in childhood are barred from consciousness, but do not cease to exist, and exert a constant pressure in the direction of consciousness. However, the content of the unconscious is only knowable to consciousness through its representation in a disguised or distorted form, by way of dreams and neurotic symptoms, as well as in slips of the tongue and jokes. The psychoanalyst seeks to interpret these conscious manifestations in order to understand the nature of the repressed.

The unconscious mind can be seen as the source of dreams and automatic thoughts (those that appear without any apparent cause), the repository of forgotten memories (that may still be accessible to consciousness at some later time), and the locus of implicit knowledge (the things that we have learned so well that we do them without thinking). Phenomena related to semi-consciousness include awakening, implicit memory, subliminal messages, trances, hypnagogia and hypnosis. While sleep, sleepwalking, dreaming, delirium and comas may signal the presence of unconscious processes, these processes are seen as symptoms rather than the unconscious mind itself.

Some critics have doubted the existence of the unconscious altogether.

Historical overview

German

The term "unconscious" (German: unbewusst) was coined by the 18th-century German Romantic philosopher Friedrich Schelling (in his System of Transcendental Idealism, ch. 6, § 3) and later introduced into English by the poet and essayist Samuel Taylor Coleridge (in his Biographia Literaria). Some rare earlier instances of the term "unconsciousness" (Unbewußtseyn) can be found in the work of the 18th-century German physician and philosopher Ernst Platner.

Vedas

Influences on thinking that originate from outside an individual's consciousness were reflected in the ancient ideas of temptation, divine inspiration, and the predominant role of the gods in affecting motives and actions. The idea of internalised unconscious processes in the mind was present in antiquity, and has been explored across a wide variety of cultures. Unconscious aspects of mentality were referred to between 2,500 and 600 BC in the Hindu texts known as the Vedas, found today in Ayurvedic medicine.

Paracelsus

Paracelsus is credited as the first to make mention of an unconscious aspect of cognition in his work Von den Krankheiten (translates as "About illnesses", 1567), and his clinical methodology created a cogent system that is regarded by some as the beginning of modern scientific psychology.

Shakespeare

William Shakespeare explored the role of the unconscious in many of his plays, without naming it as such.

Philosophy

In his work Anthropology, philosopher Immanuel Kant was one of the first to discuss the subject of unconscious ideas.

Western philosophers such as Arthur Schopenhauer, Baruch Spinoza, Gottfried Wilhelm Leibniz, Johann Gottlieb Fichte, Georg Wilhelm Friedrich Hegel, Karl Robert Eduard von Hartmann, Carl Gustav Carus, Søren Aabye Kierkegaard, Friedrich Wilhelm Nietzsche and Thomas Carlyle used the word unconscious.

In 1880 at the Sorbonne, Edmond Colsenet defended a philosophy thesis (PhD) on the unconscious. Elie Rabier and Alfred Fouillee performed syntheses of the unconscious "at a time when Freud was not interested in the concept".

Psychology

Nineteenth century

According to historian of psychology Mark Altschule, "It is difficult—or perhaps impossible—to find a nineteenth-century psychologist or psychiatrist who did not recognize unconscious cerebration as not only real but of the highest importance." In 1890, when psychoanalysis was still unheard of, William James, in his monumental treatise on psychology (The Principles of Psychology), examined the way Schopenhauer, von Hartmann, Janet, Binet and others had used the term 'unconscious' and 'subconscious.'" German psychologists, Gustav Fechner and Wilhelm Wundt, had begun to use the term in their experimental psychology, in the context of manifold, jumbled sense data that the mind organizes at an unconscious level before revealing it as a cogent totality in conscious form." Eduard von Hartmann published a book dedicated to the topic, Philosophy of the Unconscious, in 1869.

Freud

The iceberg metaphor proposed by G. T. Fechner is often used to provide a visual representation of Freud's theory that most of the human mind operates unconsciously.

Sigmund Freud and his followers developed an account of the unconscious mind. He worked with the unconscious mind to develop an explanation for mental illness.

For Freud, the unconscious is not merely that which is not conscious. He refers to that as the descriptive unconscious and it is only the starting postulate for real investigation into the psyche. He further distinguishes the unconscious from the pre-conscious: the pre-conscious is merely latent – thoughts, memories, etc. that are not present to consciousness but are capable of becoming so; the unconscious consists of psychic material that is made completely inaccessible to consciousness by the act of repression. The distinctions and inter-relationships between these three regions of the psyche—the conscious, the pre-conscious, and the unconscious—form what Freud calls the topographical model of the psyche. He later sought to respond to the perceived ambiguity of the term "unconscious" by developing what he called the structural model of the psyche, in which unconscious processes were described in terms of the id and the superego in their relation to the ego.

In the psychoanalytic view, unconscious mental processes can only be recognized through analysis of their effects in consciousness. Unconscious thoughts are not directly accessible to ordinary introspection, but they are capable of partially evading the censorship mechanism of repression in a disguised form, manifesting, for example, as dream elements or neurotic symptoms. Such symptoms are supposed to be capable of being "interpreted" during psychoanalysis, with the help of methods such as free association, dream analysis, and analysis of verbal slips and other unintentional manifestations in conscious life.

Jung

Main articles: Carl Jung and Collective unconscious

Carl Gustav Jung agreed with Freud that the unconscious is a determinant of personality, but he proposed that the unconscious be divided into two layers: the personal unconscious and the collective unconscious. The personal unconscious is a reservoir of material that was once conscious but has been forgotten or suppressed, much like Freud's notion. The collective unconscious, however, is the deepest level of the psyche, containing the accumulation of inherited psychic structures and archetypal experiences. Archetypes are not memories but energy centers or psychological functions that are apparent in the culture's use of symbols. The collective unconscious is therefore said to be inherited and contain material of an entire species rather than of an individual. The collective unconscious is, according to Jung, "[the] whole spiritual heritage of mankind's evolution, born anew in the brain structure of every individual".

In addition to the structure of the unconscious, Jung differed from Freud in that he did not believe that sexuality was at the base of all unconscious thoughts.

Dreams

Freud

The purpose of dreams, according to Freud, is to fulfill repressed wishes while simultaneously allowing the dreamer to remain asleep. The dream is a disguised fulfillment of the wish because the unconscious desire in its raw form would disturb the sleeper and can only avoid censorship by associating itself with elements that are not subject to repression. Thus Freud distinguished between the manifest content and latent content of the dream. The manifest content consists of the plot and elements of a dream as they appear to consciousness, particularly upon waking, as the dream is recalled. The latent content refers to the hidden or disguised meaning of the events and elements of the dream. It represents the unconscious psychic realities of the dreamer's current issues and childhood conflicts, the nature of which the analyst is seeking to understand through interpretation of the manifest content.

In Freud's theory, dreams are instigated by the events and thoughts of everyday life. In what he called the "dream-work", these events and thoughts, governed by the rules of language and the reality principle, become subject to the "primary process" of unconscious thought, which is governed by the pleasure principle, wish gratification and the repressed sexual scenarios of childhood. The dream-work involves a process of disguising these unconscious desires in order to preserve sleep. This process occurs primarily by means of what Freud called condensation and displacement. Condensation is the focusing of the energy of several ideas into one, and displacement is the surrender of one idea's energy to another more trivial representative. The manifest content is thus thought to be a highly significant simplification of the latent content, capable of being deciphered in the analytic process, potentially allowing conscious insight into unconscious mental activity.

Neurobiological theory of dreams

Allan Hobson and colleagues developed what they called the activation-synthesis hypothesis which proposes that dreams are simply the side effects of the neural activity in the brain that produces beta brain waves during REM sleep that are associated with wakefulness. According to this hypothesis, neurons fire periodically during sleep in the lower brain levels and thus send random signals to the cortex. The cortex then synthesizes a dream in reaction to these signals in order to try to make sense of why the brain is sending them. However, the hypothesis does not state that dreams are meaningless, it just downplays the role that emotional factors play in determining dreams.

Contemporary cognitive psychology

Research

There is an extensive body of research in contemporary cognitive psychology devoted to mental activity that is not mediated by conscious awareness. Most of this research on unconscious processes has been done in the academic tradition of the information processing paradigm. The cognitive tradition of research into unconscious processes does not rely on the clinical observations and theoretical bases of the psychoanalytic tradition; instead it is mostly data driven. Cognitive research reveals that individuals automatically register and acquire more information than they are consciously aware of or can consciously remember and report.

Much research has focused on the differences between conscious and unconscious perception. There is evidence that whether something is consciously perceived depends both on the incoming stimulus (bottom up strength) and on top-down mechanisms like attention. Recent research indicates that some unconsciously perceived information can become consciously accessible if there is cumulative evidence. Similarly, content that would normally be conscious can become unconscious through inattention (e.g. in the attentional blink) or through distracting stimuli like visual masking.

Unconscious processing of information about frequency

An extensive line of research conducted by Hasher and Zacks has demonstrated that individuals register information about the frequency of events automatically (outside conscious awareness and without engaging conscious information processing resources). Moreover, perceivers do this unintentionally, truly "automatically", regardless of the instructions they receive, and regardless of the information processing goals they have. The ability to unconsciously and relatively accurately tally the frequency of events appears to have little or no relation to the individual's age, education, intelligence, or personality. Thus it may represent one of the fundamental building blocks of human orientation in the environment and possibly the acquisition of procedural knowledge and experience, in general.

Criticism of the Freudian concept

The notion that the unconscious mind exists at all has been disputed.

Franz Brentano rejected the concept of the unconscious in his 1874 book Psychology from an Empirical Standpoint, although his rejection followed largely from his definitions of consciousness and unconsciousness.

Jean-Paul Sartre offers a critique of Freud's theory of the unconscious in Being and Nothingness, based on the claim that consciousness is essentially self-conscious. Sartre also argues that Freud's theory of repression is internally flawed. Philosopher Thomas Baldwin argues that Sartre's argument is based on a misunderstanding of Freud.

Erich Fromm contends that "The term 'the unconscious' is actually a mystification (even though one might use it for reasons of convenience, as I am guilty of doing in these pages). There is no such thing as the unconscious; there are only experiences of which we are aware, and others of which we are not aware, that is, of which we are unconscious. If I hate a man because I am afraid of him, and if I am aware of my hate but not of my fear, we may say that my hate is conscious and that my fear is unconscious; still my fear does not lie in that mysterious place: 'the' unconscious."

John Searle has offered a critique of the Freudian unconscious. He argues that the Freudian cases of shallow, consciously held mental states would be best characterized as 'repressed consciousness,' while the idea of more deeply unconscious mental states is more problematic. He contends that the very notion of a collection of "thoughts" that exist in a privileged region of the mind such that they are in principle never accessible to conscious awareness, is incoherent. This is not to imply that there are not "nonconscious" processes that form the basis of much of conscious life. Rather, Searle simply claims that to posit the existence of something that is like a "thought" in every way except for the fact that no one can ever be aware of it (can never, indeed, "think" it) is an incoherent concept. To speak of "something" as a "thought" either implies that it is being thought by a thinker or that it could be thought by a thinker. Processes that are not causally related to the phenomenon called thinking are more appropriately called the nonconscious processes of the brain.

Other critics of the Freudian unconscious include David Stannard, Richard Webster, Ethan Watters, Richard Ofshe, and Eric Thomas Weber.

Some scientific researchers proposed the existence of unconscious mechanisms that are very different from the Freudian ones. They speak of a "cognitive unconscious" (John Kihlstrom), an "adaptive unconscious" (Timothy Wilson), or a "dumb unconscious" (Loftus and Klinger), which executes automatic processes but lacks the complex mechanisms of repression and symbolic return of the repressed, and the "deep unconscious system" of Robert Langs.

In modern cognitive psychology, many researchers have sought to strip the notion of the unconscious from its Freudian heritage, and alternative terms such as "implicit" or "automatic" have been used. These traditions emphasize the degree to which cognitive processing happens outside the scope of cognitive awareness, and show that things we are unaware of can nonetheless influence other cognitive processes as well as behavior. Active research traditions related to the unconscious include implicit memory (for example, priming), and Pawel Lewicki's nonconscious acquisition of knowledge.

Lucid dream

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lucid_dream 

In the psychology subfield of oneirology, a lucid dream is a type of dream wherein the dreamer realizes that they are dreaming during their dream. The capacity to have and sustain lucid dreams is a trainable cognitive skill. During a lucid dream, the dreamer may gain some amount of volitional control over the dream characters, narrative, or environment, although this control of dream content is not the salient feature of lucid dreaming. An important distinction is that lucid dreaming is a distinct type of dream from other types of dreams such as prelucid dreams and vivid dreams, although prelucid dreams are a precursor to lucid dreams, and lucid dreams are often accompanied with enhanced dream vividness. Lucid dreams are also a distinct state from other lucid boundary sleep states such as lucid hypnagogia or lucid hypnopompia.

In formal psychology, lucid dreaming has been studied and reported for many years. Prominent figures from ancient to modern times have been fascinated by lucid dreams and have sought ways to better understand their causes and purpose. Many different theories have emerged as a result of scientific research on the subject. Further developments in psychological research have pointed to ways in which this form of dreaming may be utilized as a therapeutic technique.

The term lucid dream was coined by Dutch author and psychiatrist Frederik van Eeden in his 1913 article A Study of Dreams, though descriptions of dreamers being aware that they are dreaming predate the article. Psychologist Stephen LaBerge is widely considered the progenitor and leading pioneer of modern lucid dreaming research. He is the founder of the Lucidity Institute at Stanford University.

Definition

Paul Tholey laid the epistemological basis for the research of lucid dreams, proposing seven different conditions of clarity that a dream must fulfill to be defined as a lucid dream:

1. Awareness of the dream state (orientation)
2. Awareness of the capacity to make decisions
3. Awareness of memory functions
4. Awareness of self
5. Awareness of the dream environment
6. Awareness of the meaning of the dream
7. Awareness of concentration and focus (the subjective clarity of that state)

Later, in 1992, a study by Deirdre Barrett examined whether lucid dreams contained four "corollaries" of lucidity:

1. The dreamer is aware that they are dreaming
2. They are aware that actions will not carry over after waking
3. Physical laws need not apply in the dream
4. The dreamer has a clear memory of the waking world

Barrett found that less than a quarter of lucidity accounts exhibited all four.

Subsequently, Stephen LaBerge studied the prevalence among lucid dreams of the ability to control the dream scenario, and found that while dream control and dream awareness are correlated, neither requires the other. LaBerge found dreams that exhibit one clearly without the capacity for the other. He also found dreams where, although the dreamer is lucid and aware they could exercise control, they choose simply to observe.

History

Eastern

The practice of lucid dreaming is central to both the ancient Indian Hindu practice of Yoga nidra and the Tibetan Buddhist practice of dream Yoga. The cultivation of such awareness was a common practice among early Buddhists.

Western

Early references to the phenomenon are also found in ancient Greek writing. For example, the philosopher Aristotle wrote: "often when one is asleep, there is something in consciousness which declares that what then presents itself is but a dream." Meanwhile, the physician Galen of Pergamon used lucid dreams as a form of therapy. In addition, a letter written by Saint Augustine of Hippo in AD 415 tells the story of a dreamer, Doctor Gennadius, and refers to lucid dreaming.

Philosopher and physician Sir Thomas Browne (1605–1682) was fascinated by dreams and described his own ability to lucid dream in his Religio Medici, stating: "...yet in one dream I can compose a whole Comedy, behold the action, apprehend the jests and laugh my self awake at the conceits thereof."

Samuel Pepys, in his diary entry for 15 August 1665, records a dream, stating: "I had my Lady Castlemayne in my arms and was admitted to use all the dalliance I desired with her, and then dreamt that this could not be awake, but that it was only a dream."

In 1867, the French sinologist Marie-Jean-Léon, Marquis d'Hervey de Saint Denys anonymously published Les Rêves et Les Moyens de Les Diriger; Observations Pratiques ("Dreams and the ways to direct them; practical observations"), in which he describes his own experiences of lucid dreaming, and proposes that it is possible for anyone to learn to dream consciously.

Frederik van Eeden (left) and Marquis d'Hervey de Saint Denys (right), early researchers of lucid dreaming

In 1913, Dutch psychiatrist and writer Frederik (Willem) van Eeden (1860–1932) coined the term "lucid dream" in an article entitled "A Study of Dreams".

Some have suggested that the term is a misnomer because Van Eeden was referring to a phenomenon more specific than a lucid dream. Van Eeden intended the term lucid to denote "having insight", as in the phrase a lucid interval applied to someone in temporary remission from a psychosis, rather than as a reference to the perceptual quality of the experience, which may or may not be clear and vivid.

Skill mastery

Clinical psychologist, Kristen LaMarca outlined four stages towards mastering the skill of using lucid dreaming:

Lucid Dreaming Skill Levels

Stage	Title	Description	Rarity
1	Beginner	The practitioner may have no recollection of ever having a lucid dream, and perhaps has at most experienced only brief moments of lucidity.	Common
2	Experienced	An experienced lucid dreaming practitioner wields an increased ability of dream control and capacity to execute pre-intended actions. However, there are still aspects of lucid dream practice about employing lucidity productively that are yet to be honed. One's understanding of accessing and maintaining dream lucidity deepen as one has more lucid dreams.	Less common
3	Proficient	A proficient lucid dreamer is marked by a deliberate ability to accomplish intended actions in lucid dreams, along with knowledge of the best actions for given dream scenarios. The proficient lucid dream practitioner's practice is well-planned, drawing upon a broad skill set facilitating flexible oneironautic exploration, which can include contemplative practices or athletic motor skill training. This level of skill adequacy is not necessary to develop a fulfilling lucid dream practice.	Uncommon
4	Expert	Expertise in lucid dream skill is accompanied by normalization of greater intensity of lucidity during lucid dreams. LaMarca writes that the expert's practice is "characterized by at least a decade of intense dedication, long training hours, and mentorship by other more advanced experts." Spiritual figures, such Tibetan Buddhist masters, tend to display the highest order of mastery.	Extremely rare

Progression along the skill levels is akin to a maturity in the development of the practitioner's discipline, methodology and application.

Cognitive science

In 1968, Celia Green analyzed the main characteristics of such dreams, reviewing previously published literature on the subject and incorporating new data from participants of her own. She concluded that lucid dreams were a category of experience quite distinct from ordinary dreams and said they were associated with rapid eye movement sleep (REM sleep). Green was also the first to link lucid dreams to the phenomenon of false awakenings, which has since been corroborated by more recent studies.

In 1973, the National Institute of Mental Health reported that researchers at the University of California, San Francisco, were able to train sleeping subjects to recognize they were in REM dreaming and indicate this by pressing micro switches on their thumbs. Using tones and mild shocks as cues, the experiments showed that the subjects were able to signal knowledge of their various sleep stages, including dreaming.

In 1975, Dr. Keith Hearne had the idea to exploit the nature of rapid eye movements (REM) to allow a dreamer to send a message directly from dreams to the waking world. Working with an experienced lucid dreamer (Alan Worsley), he eventually succeeded in recording (via the use of an electrooculogram or EOG) a pre-defined set of eye movements signaled from within Worsley's lucid dream. This occurred at around 8 am on the morning of April 12, 1975. Hearne's EOG experiment was formally recognized through publication in the journal for The Society for Psychical Research. Lucid dreaming was subsequently researched by asking dreamers to perform pre-determined physical responses while experiencing a dream, including eye movement signals.

In 1980, Stephen LaBerge at Stanford University developed such techniques as part of his doctoral dissertation. In 1985, LaBerge performed a pilot study that showed that time perception while counting during a lucid dream is about the same as during waking life. Lucid dreamers counted out ten seconds while dreaming, signaling the start and the end of the count with a pre-arranged eye signal measured with electrooculogram recording. LaBerge's results were confirmed by German researchers D. Erlacher and M. Schredl in 2004.

In a further study by Stephen LaBerge, four subjects were compared, either singing or counting while dreaming. LaBerge found that the right hemisphere was more active during singing and the left hemisphere was more active during counting.

Neuroscientist J. Allan Hobson has hypothesized what might be occurring in the brain while lucid. The first step to lucid dreaming is recognizing that one is dreaming. This recognition might occur in the dorsolateral prefrontal cortex, which is one of the few areas deactivated during REM sleep and where working memory occurs. Once this area is activated and the recognition of dreaming occurs, the dreamer must be cautious to let the dream continue, but be conscious enough to remember that it is a dream. While maintaining this balance, the amygdala and parahippocampal cortex might be less intensely activated. To continue the intensity of the dream hallucinations, it is expected the pons and the parieto-occipital junction stay active.

Using electroencephalography (EEG) and other polysomnographical measurements, LaBerge and others have shown that lucid dreams begin in the rapid eye movement (REM) stage of sleep. LaBerge also proposes that there are higher amounts of beta-1 frequency band (13–19 Hz) brain wave activity experienced by lucid dreamers, hence there is an increased amount of activity in the parietal lobes making lucid dreaming a conscious process.

Paul Tholey, a German Gestalt psychologist and a professor of psychology and sports science, originally studied dreams in order to resolve the question of whether one dreams in colour or black and white. In his phenomenological research, he outlined an epistemological frame using critical realism. Tholey instructed his subjects to continuously suspect waking life to be a dream, in order that such a habit would manifest itself during dreams. He called this technique for inducing lucid dreams the Reflexionstechnik (reflection technique). Subjects learned to have such lucid dreams; they observed their dream content and reported it soon after awakening. Tholey could examine the cognitive abilities of dream figures. Nine trained lucid dreamers were directed to set other dream figures arithmetic and verbal tasks during lucid dreaming. Dream figures who agreed to perform the tasks proved more successful in verbal than in arithmetic tasks. Tholey discussed his scientific results with Stephen LaBerge, who has a similar approach.

A study was conducted by Stephen LaBerge and other scientists to see if it were possible to attain the ability to lucid dream through a drug. In 2018, galantamine was given to 121 patients in a double-blind, placebo-controlled trial, the only one of its kind. Some participants found as much as a 42 percent increase in their ability to lucid dream, compared to self-reports from the past six months, and ten people experienced a lucid dream for the first time. It is theorized that galantamine allows acetylcholine to build up, leading to greater recollection and awareness during dreaming.

Two-way communication

Graphical abstract of "Real-time dialogue between experimenters and dreamers during REM sleep"

Teams of cognitive scientists have established real-time two-way communication with people undergoing a lucid dream. During dreaming, they were able to consciously communicate with experimenters via eye movements or facial muscle signals, were able to comprehend complex questions and use working memory. Such interactive lucid dreaming could be a new approach for the scientific exploration of the dream state and could have applications for learning and creativity.

Alternative theories

Other researchers suggest that lucid dreaming is not a state of sleep, but of brief wakefulness, or "micro-awakening". Experiments by Stephen LaBerge used "perception of the outside world" as a criterion for wakefulness while studying lucid dreamers, and their sleep state was corroborated with physiological measurements. LaBerge's subjects experienced their lucid dream while in a state of REM, which critics felt may mean that the subjects are fully awake. J. Allen Hobson responded that lucid dreaming must be a state of both waking and dreaming.

Philosopher Norman Malcolm was a proponent of dream skepticism. He has argued against the possibility of checking the accuracy of dream reports, pointing out that "the only criterion of the truth of a statement that someone has had a certain dream is, essentially, his saying so." Yet dream reports are not the only evidence that some inner drama is being played out during REM sleep. Electromyography on speech and body muscles has demonstrated the sleeping body covertly walking, gesturing and talking while in REM.

Prevalence and frequency

In 2016, a meta-analytic study by David Saunders and colleagues on 34 lucid dreaming studies, taken from a period of 50 years, demonstrated that 55% of a pooled sample of 24,282 people claimed to have experienced lucid dreams at least once or more in their lifetime. Furthermore, for those that stated they did experience lucid dreams, approximately 23% reported to experience them on a regular basis, as often as once a month or more. In a 2004 study on lucid dream frequency and personality, a moderate correlation between nightmare frequency and frequency of lucid dreaming was demonstrated. Some lucid dreamers also reported that nightmares are a trigger for dream lucidity. Previous studies have reported that lucid dreaming is more common among adolescents than adults.

A 2015 study by Julian Mutz and Amir-Homayoun Javadi showed that people who had practiced meditation for a long time tended to have more lucid dreams. The authors claimed that "Lucid dreaming is a hybrid state of consciousness with features of both waking and dreaming" in a review they published in Neuroscience of Consciousness in 2017.

Mutz and Javadi found that during lucid dreaming, there is an increase in activity of the dorsolateral prefrontal cortex, the bilateral frontopolar prefrontal cortex, the precuneus, the inferior parietal lobules, and the supramarginal gyrus. All are brain functions related to higher cognitive functions, including working memory, planning, and self-consciousness. The researchers also found that during a lucid dream, "levels of self-determination" were similar to those that people experienced during states of wakefulness. They also found that lucid dreamers can only control limited aspects of their dream at once.

Mutz and Javadi also have stated that by studying lucid dreaming further, scientists could learn more about various types of consciousness, which happen to be less easy to separate and research at other times.

Suggested applications

Treating nightmares

It has been suggested that those who suffer from nightmares could benefit from the ability to be aware they are indeed dreaming. A pilot study performed in 2006 showed that lucid dreaming therapy treatment was successful in reducing nightmare frequency. This treatment consisted of exposure to the idea, mastery of the technique, and lucidity exercises. It was not clear what aspects of the treatment were responsible for the success of overcoming nightmares, though the treatment as a whole was said to be successful.

Australian psychologist Milan Colic has explored the application of principles from narrative therapy to clients' lucid dreams, to reduce the impact not only of nightmares during sleep but also depression, self-mutilation, and other problems in waking life. Colic found that therapeutic conversations could reduce the distressing content of dreams, while understandings about life—and even characters—from lucid dreams could be applied to their lives with marked therapeutic benefits.

Psychotherapists have applied lucid dreaming as a part of therapy. Studies have shown that, by inducing a lucid dream, recurrent nightmares can be alleviated. It is unclear whether this alleviation is due to lucidity or the ability to alter the dream itself. A 2006 study performed by Victor Spoormaker and Van den Bout evaluated the validity of lucid dreaming treatment (LDT) in chronic nightmare sufferers. LDT is composed of exposure, mastery and lucidity exercises. Results of lucid dreaming treatment revealed that the nightmare frequency of the treatment groups had decreased. In another study, Spoormaker, Van den Bout, and Meijer (2003) investigated lucid dreaming treatment for nightmares by testing eight subjects who received a one-hour individual session, which consisted of lucid dreaming exercises. The results of the study revealed that the nightmare frequency had decreased and the sleep quality had slightly increased.

Holzinger, Klösch, and Saletu managed a psychotherapy study under the working name of 'Cognition during dreaming—a therapeutic intervention in nightmares', which included 40 subjects, men and women, 18–50 years old, whose life quality was significantly altered by nightmares. The test subjects were administered Gestalt group therapy, and 24 of them were also taught to enter the state of lucid dreaming by Holzinger. This was purposefully taught in order to change the course of their nightmares. The subjects then reported the diminishment of their nightmare prevalence from 2–3 times a week to 2–3 times per month.

Creativity

In her book The Committee of Sleep, Deirdre Barrett describes how some experienced lucid dreamers have learned to remember specific practical goals such as artists looking for inspiration seeking a show of their own work once they become lucid or computer programmers looking for a screen with their desired code. However, most of these dreamers had many experiences of failing to recall waking objectives before gaining this level of control.

Exploring the World of Lucid Dreaming by Stephen LaBerge and Howard Rheingold (1990) discusses creativity within dreams and lucid dreams, including testimonials from a number of people who claim they have used the practice of lucid dreaming to help them solve a number of creative issues, from an aspiring parent thinking of potential baby names to a surgeon practicing surgical techniques. The authors discuss how creativity in dreams could stem from "conscious access to the contents of our unconscious minds"; access to "tacit knowledge"—the things we know but can't explain, or things we know but are unaware that we know.

The Dreams Behind the Music book by Craig Webb (2016) details lucid dreams of a number of musical artists, including how they are able not just to hear, but also compose, mix, arrange, practice, and perform music while conscious within their dreams.

Risks

Though lucid dreaming can be beneficial to a number of aspects of life, some risks have been suggested. Those struggling with certain mental illnesses could find it hard to tell the difference between reality and the lucid dream (psychosis).

A very small percentage of people may experience sleep paralysis, which can sometimes be confused with lucid dreaming. Although from the outside, both seem to be quite similar, there are a few distinct differences that can help differentiate them. A person usually experiences sleep paralysis when they partially wake up in REM atonia, a state in which said person is partially paralyzed and cannot move their limbs. When in sleep paralysis, people may also experience hallucinations. Although said hallucinations cannot cause physical damage, they may still be frightening. There are three common types of hallucinations: an intruder in the same room, a crushing feeling on one's chest or back, and a feeling of flying or levitating. About 7.6% of the general population have experienced sleep paralysis at least once. Exiting sleep paralysis to a waking state can be achieved by intently focusing on a part of the body, such as a finger, and wiggling it, then continuing the action of moving to the hand, the arm, and so on, until the person is fully awake.

Long-term risks with lucid dreaming have not been extensively studied, although many people have reported lucid dreaming for many years without any adverse effects. In 2018, researchers at the Wisconsin Institute for Sleep and Consciousness conducted a study that concluded individuals who lucid dream more frequently have a more active and well-connected prefrontal cortex.

Self-reference

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Self-reference 

The ancient symbol Ouroboros, a dragon that continually consumes itself, denotes self-reference.

Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.

In natural or formal languages, self-reference occurs when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding.

In philosophy, self-reference also refers to the ability of a subject to speak of or refer to itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English.

Self-reference is studied and has applications in mathematics, philosophy, computer programming, second-order cybernetics, and linguistics, as well as in humor. Self-referential statements are sometimes paradoxical, and can also be considered recursive.

In logic, mathematics and computing

In classical philosophy, paradoxes were created by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretans are liars' when uttered by an ancient Greek Cretan was one of the first recorded versions. Contemporary philosophy sometimes employs the same technique to demonstrate that a supposed concept is meaningless or ill-defined.

In mathematics and computability theory, self-reference (also known as impredicativity) is the key concept in proving limitations of many systems. Gödel's theorem uses it to show that no formal consistent system of mathematics can ever contain all possible mathematical truths, because it cannot prove some truths about its own structure. The halting problem equivalent, in computation theory, shows that there is always some task that a computer cannot perform, namely reasoning about itself. These proofs relate to a long tradition of mathematical paradoxes such as Russell's paradox and Berry's paradox, and ultimately to classical philosophical paradoxes.

In game theory, undefined behaviors can occur where two players must model each other's mental states and behaviors, leading to infinite regress.

In computer programming, self-reference occurs in reflection, where a program can read or modify its own instructions like any other data. Numerous programming languages support reflection to some extent with varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation) in functional programming, where a code structure refers back to itself during computation. 'Taming' self-reference from potentially paradoxical concepts into well-behaved recursions has been one of the great successes of computer science, and is now used routinely in, for example, writing compilers using the 'meta-language' ML. Using a compiler to compile itself is known as bootstrapping. Self-modifying code is possible to write (programs which operate on themselves), both with assembler and with functional languages such as Lisp, but is generally discouraged in real-world programming. Computing hardware makes fundamental use of self-reference in flip-flops, the basic units of digital memory, which convert potentially paradoxical logical self-relations into memory by expanding their terms over time. Thinking in terms of self-reference is a pervasive part of programmer culture, with many programs and acronyms named self-referentially as a form of humor, such as GNU ('GNU's not Unix') and PINE ('Pine is not Elm'). The GNU Hurd is named for a pair of mutually self-referential acronyms.

Tupper's self-referential formula is a mathematical curiosity which plots an image of its own formula.

In art

Drawloom, with drawboy above to control the harnesses, woven as a repeating pattern in an early-1800s piece of Japanese silk. The silk illustrates the means by which it was produced.

A self-referencing work of graffiti apologizing for its own existence

Self-referential graffiti. The painter drawn on a wall erases his own graffiti, and may be erased himself by the next facade cleaner.

Self-reference occurs in literature and film when an author refers to his or her own work in the context of the work itself. Examples include Miguel de Cervantes' Don Quixote, Shakespeare's A Midsummer Night's Dream, The Tempest and Twelfth Night, Denis Diderot's Jacques le fataliste et son maître, Italo Calvino's If on a winter's night a traveler, many stories by Nikolai Gogol, Lost in the Funhouse by John Barth, Luigi Pirandello's Six Characters in Search of an Author, Federico Fellini's 8½ and Bryan Forbes's The L-Shaped Room. Speculative fiction writer Samuel R. Delany makes use of this in his novels Nova and Dhalgren. In the former, Katin (a space-faring novelist) is wary of a long-standing curse wherein a novelist dies before completing any given work. Nova ends mid-sentence, thus lending credence to the curse and the realization that the novelist is the author of the story; likewise, throughout Dhalgren, Delany has a protagonist simply named The Kid (or Kidd, in some sections), whose life and work are mirror images of themselves and of the novel itself. In the sci-fi spoof film Spaceballs, Director Mel Brooks includes a scene wherein the evil characters are viewing a VHS copy of their own story, which shows them watching themselves "watching themselves", ad infinitum. Perhaps the earliest example is in Homer's Iliad, where Helen of Troy laments: "for generations still unborn/we will live in song" (appearing in the song itself).

Self-reference in art is closely related to the concepts of breaking the fourth wall and meta-reference, which often involve self-reference. The short stories of Jorge Luis Borges play with self-reference and related paradoxes in many ways. Samuel Beckett's Krapp's Last Tape consists entirely of the protagonist listening to and making recordings of himself, mostly about other recordings. During the 1990s and 2000s filmic self-reference was a popular part of the rubber reality movement, notably in Charlie Kaufman's films Being John Malkovich and Adaptation, the latter pushing the concept arguably to its breaking point as it attempts to portray its own creation, in a dramatized version of the Droste effect.

Various creation myths invoke self-reference to solve the problem of what created the creator. For example, the Egyptian creation myth has a god swallowing his own semen to create himself. The Ouroboros is a mythical dragon which eats itself.

The Quran includes numerous instances of self-referentiality.

The surrealist painter René Magritte is famous for his self-referential works. His painting The Treachery of Images, includes the words "this is not a pipe", the truth of which depends entirely on whether the word ceci (in English, "this") refers to the pipe depicted—or to the painting or the word or sentence itself. M.C. Escher's art also contains many self-referential concepts such as hands drawing themselves.

In language

A word that describes itself is called an autological word (or autonym). This generally applies to adjectives, for example sesquipedalian (i.e. "sesquipedalian" is a sesquipedalian word), but can also apply to other parts of speech, such as TLA, as a three-letter abbreviation for "three-letter abbreviation".

A sentence which inventories its own letters and punctuation marks is called an autogram.

There is a special case of meta-sentence in which the content of the sentence in the metalanguage and the content of the sentence in the object language are the same. Such a sentence is referring to itself. However some meta-sentences of this type can lead to paradoxes. "This is a sentence." can be considered to be a self-referential meta-sentence which is obviously true. However "This sentence is false" is a meta-sentence which leads to a self-referential paradox. Such sentences can lead to problems, for example, in law, where statements bringing laws into existence can contradict one another or themselves. Kurt Gödel claimed to have found such a loophole in the United States Constitution at his citizenship ceremony.

Self-reference occasionally occurs in the media when it is required to write about itself, for example the BBC reporting on job cuts at the BBC. Notable encyclopedias may be required to feature articles about themselves, such as Wikipedia's article on Wikipedia.

Fumblerules are a list of rules of good grammar and writing, demonstrated through sentences that violate those very rules, such as "Avoid cliches like the plague" and "Don't use no double negatives". The term was coined in a published list of such rules by William Safire.

Circular definition is a type of self-reference in which the definition of a term or concept includes the term or concept itself, either explicitly or implicitly. Circular definitions are considered fallacious because they only define a term in terms of itself. This type of self-reference may be useful in argumentation, but can result in a lack of clarity in communication.

The adverb "hereby" is used in a self-referential way, for example in the statement "I hereby declare you husband and wife."

In popular culture

• Douglas Hofstadter's books, especially Metamagical Themas and Gödel, Escher, Bach, play with many self-referential concepts and were highly influential in bringing them into mainstream intellectual culture during the 1980s. Hofstadter's law, which specifies that "It always takes longer than you expect, even when you take into account Hofstadter's Law" is an example of a self-referencing adage. Hofstadter also suggested the concept of a 'Reviews of this book', a book containing only reviews of itself, which has since been implemented using wikis and other technologies. Hofstadter's 'strange loop' metaphysics attempts to map consciousness onto self-reference, but is a minority position in philosophy of mind.
• The subgenre of "recursive science fiction" or metafiction is now so extensive that it has fostered a fan-maintained bibliography at the New England Science Fiction Association's website; some of it is about science-fiction fandom, some about science fiction and its authors.

In law

Several constitutions contain self-referential clauses defining how the constitution itself may be amended. An example is Article Five of the United States Constitution.