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Survival of the fittest
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For other uses, see Survival of the fittest (disambiguation).
Herbert Spencer coined the phrase "survival of the fittest".
"Survival of the fittest" is a phrase that originated from Darwinian evolutionary theory as a way of describing the mechanism of natural selection. The biological concept of fitness is defined as reproductive success. In Darwinian terms the phrase is best understood as "Survival of the form that will leave the most copies of itself in successive generations."
Herbert Spencer first used the phrase, after reading Charles Darwin's On the Origin of Species, in his Principles of Biology (1864), in which he drew parallels between his own economic theories and Darwin's biological ones: "This survival of the fittest, which I have here sought to express in mechanical terms, is that which Mr. Darwin has called 'natural selection', or the preservation of favoured races in the struggle for life."[1]
Darwin responded positively to Alfred Russel Wallace's suggestion of using Spencer's new phrase "survival of the fittest" as an alternative to "natural selection", and adopted the phrase in The Variation of Animals and Plants under Domestication published in 1868.[1][2] In On the Origin of Species, he introduced the phrase in the fifth edition published in 1869,[3][4] intending it to mean "better designed for an immediate, local environment".[5][6]
Contents
[hide]
1 History of the phrase
2 Critiquing the phrase “survival of the fittest”
3 Interpreted as expressing a biological theory
4 Interpreted as expressing a moral theory
4.1 Social Darwinists
4.2 Creationists
4.3 Anarchists
5 Tautology
6 See also
7 Notes
8 References
9 External links
9.1 Origins of the phrase
9.2 Tautology links
9.3 Morality link
9.4 Kropotkin: Mutual Aid
History of the phrase[edit]
Herbert Spencer first used the phrase – after reading Charles Darwin's On the Origin of Species – in his Principles of Biology of 1864[7] in which he drew parallels between his economic theories and Darwin's biological, evolutionary ones, writing, "This survival of the fittest, which I have here sought to express in mechanical terms, is that which Mr. Darwin has called 'natural selection', or the preservation of favored races in the struggle for life."[1]
In July 1866 Alfred Russel Wallace wrote to Darwin about readers thinking that the phrase "natural selection" personified nature as "selecting", and said this misconception could be avoided "by adopting Spencer's term" Survival of the fittest. Darwin promptly replied that Wallace's letter was "as clear as daylight. I fully agree with all that you say on the advantages of H. Spencer's excellent expression of 'the survival of the fittest'. This however had not occurred to me till reading your letter. It is, however, a great objection to this term that it cannot be used as a substantive governing a verb". Had he received the letter two months earlier, he would have worked the phrase into the fourth edition of the Origin which was then being printed, and he would use it in his "next book on Domestic Animals etc.".[1]
Darwin wrote on page 6 of The Variation of Animals and Plants under Domestication published in 1868, "This preservation, during the battle for life, of varieties which possess any advantage in structure, constitution, or instinct, I have called Natural Selection; and Mr. Herbert Spencer has well expressed the same idea by the Survival of the Fittest. The term "natural selection" is in some respects a bad one, as it seems to imply conscious choice; but this will be disregarded after a little familiarity". He defended his analogy as similar to language used in chemistry, and to astronomers depicting the "attraction of gravity as ruling the movements of the planets", or the way in which "agriculturists speak of man making domestic races by his power of selection". He had "often personified the word Nature; for I have found it difficult to avoid this ambiguity; but I mean by nature only the aggregate action and product of many natural laws,—and by laws only the ascertained sequence of events."[2]
In the first four editions of On the Origin of Species, Darwin had used the phrase "natural selection".[8] In Chapter 4 of the 5th edition of The Origin published in 1869,[3] Darwin implies again the synonym: "Natural Selection, or the Survival of the Fittest".[4] By "fittest" Darwin meant "better adapted for the immediate, local environment", not the common modern meaning of "in the best physical shape" (think of a puzzle piece, not an athlete).[5] In the introduction he gave full credit to Spencer, writing "I have called this principle, by which each slight variation, if useful, is preserved, by the term Natural Selection, in order to mark its relation to man's power of selection. But the expression often used by Mr. Herbert Spencer of the Survival of the Fittest is more accurate, and is sometimes equally convenient."[9]
In The Man Versus The State, Spencer used the phrase in a postscript to justify a plausible explanation of how his theories would not be adopted by "societies of militant type". He uses the term in the context of societies at war, and the form of his reference suggests that he is applying a general principle.[10]
"Thus by survival of the fittest, the militant type of society becomes characterized by profound confidence in the governing power, joined with a loyalty causing submission to it in all matters whatever".[11]
Though Spencer’s conception of organic evolution is commonly interpreted as a form of Lamarckism,[a] Herbert Spencer is sometimes credited with inaugurating Social Darwinism. The phrase "survival of the fittest" has become widely used in popular literature as a catchphrase for any topic related or analogous to evolution and natural selection. It has thus been applied to principles of unrestrained competition, and it has been used extensively by both proponents and opponents of Social Darwinism.[citation needed]
Evolutionary biologists criticise the manner in which the term is used by non-scientists and the connotations that have grown around the term in popular culture. The phrase also does not help in conveying the complex nature of natural selection, so modern biologists prefer and almost exclusively use the term natural selection. The biological concept of fitness refers to reproductive success, as opposed to survival, and is not explicit in the specific ways in which organisms can be more "fit" (increase reproductive success) as having phenotypic characteristics that enhance survival and reproduction (which was the meaning that Spencer had in mind).[citation needed]
Critiquing the phrase “survival of the fittest”[edit]
While the phrase "survival of the fittest” is often used to refer to “natural selection”, it is avoided by modern biologists, because the phrase can be misleading. For example, “survival” is only one aspect of selection, and not always the most important. Another problem is that the word “fit” is frequently confused with a state of physical fitness. In the evolutionary meaning “fitness” is the rate of reproductive output among a class of genetic variants.[13]
Interpreted as expressing a biological theory[edit]
The phrase can also be interpreted to express a theory or hypothesis: that "fit" as opposed to "unfit" individuals or species, in some sense of "fit", will survive some test.
Interpretations of the phrase as expressing a theory are in danger of being tautological, meaning roughly "those with a propensity to survive have a propensity to survive"; to have content the theory must use a concept of fitness that is independent of that of survival.[5][14]
Interpreted as a theory of species survival, the theory that the fittest species survive is undermined by evidence that while direct competition is observed between individuals, populations and species, there is little evidence that competition has been the driving force in the evolution of large groups such as, for example, amphibians, reptiles, and mammals. Instead, these groups have evolved by expanding into empty ecological niches.[15] In the punctuated equilibrium model of environmental and biological change, the factor determining survival is often not superiority over another in competition but ability to survive dramatic changes in environmental conditions, such as after a meteor impact energetic enough to greatly change the environment globally. The main land dwelling animals to survive the K-Pg impact 66 million years ago had the ability to live in underground tunnels, for example.
In 2010 Sahney et al. argued that there is little evidence that intrinsic, biological factors such as competition have been the driving force in the evolution of large groups. Instead, they cited extrinsic, abiotic factors such as expansion as the driving factor on a large evolutionary scale. The rise of dominant groups such as amphibians, reptiles, mammals and birds occurred by opportunistic expansion into empty ecological niches and the extinction of groups happened due to large shifts in the abiotic environment.[15]
Interpreted as expressing a moral theory[edit]
Social Darwinists[edit]
It has been claimed that "the survival of the fittest" theory in biology was interpreted by late 19th century capitalists as "an ethical precept that sanctioned cut-throat economic competition" and led to the advent of the theory of "social Darwinism" which was used to justify laissez-faire economics, war and racism. However, these ideas predate and commonly contradict Darwin's ideas, and indeed their proponents rarely invoked Darwin in support.[citation needed] The term "social Darwinism" referring to capitalist ideologies was introduced as a term of abuse by Richard Hofstadter's Social Darwinism in American Thought published in 1944.[16][17]
Creationists[edit]
Critics of theories of evolution have argued that "survival of the fittest" provides a justification for behaviour that undermines moral standards by letting the strong set standards of justice to the detriment of the weak.[18] However, any use of evolutionary descriptions to set moral standards would be a naturalistic fallacy (or more specifically the is–ought problem), as prescriptive moral statements cannot be derived from purely descriptive premises. Describing how things are does not imply that things ought to be that way. It is also suggested that "survival of the fittest" implies treating the weak badly, even though in some cases of good social behaviour – co-operating with others and treating them well – might improve evolutionary fitness.[16][19]
Anarchists[edit]
Russian anarchist Peter Kropotkin viewed the concept of "survival of the fittest" as supporting co-operation rather than competition. In his book Mutual Aid: A Factor of Evolution he set out his analysis leading to the conclusion that the fittest was not necessarily the best at competing individually, but often the community made up of those best at working together. He concluded that
In the animal world we have seen that the vast majority of species live in societies, and that they find in association the best arms for the struggle for life: understood, of course, in its wide Darwinian sense — not as a struggle for the sheer means of existence, but as a struggle against all natural conditions unfavourable to the species. The animal species, in which individual struggle has been reduced to its narrowest limits, and the practice of mutual aid has attained the greatest development, are invariably the most numerous, the most prosperous, and the most open to further progress.
Applying this concept to human society, Kropotkin presented mutual aid as one of the dominant factors of evolution, the other being self-assertion, and concluded that
In the practice of mutual aid, which we can retrace to the earliest beginnings of evolution, we thus find the positive and undoubted origin of our ethical conceptions; and we can affirm that in the ethical progress of man, mutual support not mutual struggle – has had the leading part. In its wide extension, even at the present time, we also see the best guarantee of a still loftier evolution of our race.
Tautology[edit]
"Survival of the fittest" is sometimes claimed to be a tautology.[20] The reasoning is that if one takes the term "fit" to mean "endowed with phenotypic characteristics which improve chances of survival and reproduction" (which is roughly how Spencer understood it), then "survival of the fittest" can simply be rewritten as "survival of those who are better equipped for surviving". Furthermore, the expression does become a tautology if one uses the most widely accepted definition of "fitness" in modern biology, namely reproductive success itself (rather than any set of characters conducive to this reproductive success). This reasoning is sometimes used to claim that Darwin's entire theory of evolution by natural selection is fundamentally tautological, and therefore devoid of any explanatory power.[20]
However, the expression "survival of the fittest" (taken on its own and out of context) gives a very incomplete account of the mechanism of natural selection. The reason is that it does not mention a key requirement for natural selection, namely the requirement of heritability. It is true that the phrase "survival of the fittest", in and by itself, is a tautology if fitness is defined by survival and reproduction. Natural selection is the portion of variation in reproductive success that is caused by heritable characters (see the article on natural selection).[20]
If certain heritable characters increase or decrease the chances of survival and reproduction of their bearers, then it follows mechanically (by definition of "heritable") that those characters that improve survival and reproduction will increase in frequency over generations. This is precisely what is called "evolution by natural selection." On the other hand, if the characters which lead to differential reproductive success are not heritable, then no meaningful evolution will occur, "survival of the fittest" or not: if improvement in reproductive success is caused by traits that are not heritable, then there is no reason why these traits should increase in frequency over generations. In other words, natural selection does not simply state that "survivors survive" or "reproducers reproduce"; rather, it states that "survivors survive, reproduce and therefore propagate any heritable characters which have affected their survival and reproductive success". This statement is not tautological: it hinges on the testable hypothesis that such fitness-impacting heritable variations actually exist (a hypothesis that has been amply confirmed.)[20]
Momme von Sydow suggested further definitions of 'survival of the fittest' that may yield a testable meaning in biology and also in other areas where Darwinian processes have been influential. However, much care would be needed to disentangle tautological from testable aspects. Moreover, an "implicit shifting between a testable and an untestable interpretation can be an illicit tactic to immunize natural selection [...] while conveying the impression that one is concerned with testable hypotheses."[14][21]
Skeptic Society founder and Skeptic magazine publisher Michael Shermer addresses the tautology problem in his 1997 book, Why People Believe Weird Things, in which he points out that although tautologies are sometimes the beginning of science, they are never the end, and that scientific principles like natural selection are testable and falsifiable by virtue of their predictive power. Shermer points out, as an example, that population genetics accurately demonstrate when natural selection will and will not effect change on a population. Shermer hypothesizes that if hominid fossils were found in the same geological strata as trilobites, it would be evidence against natural selection.[22]
See also[edit]
Age of the Earth
Anarchism
Altruism
Robert Boyle
Capitalism
Darwinian puzzle
Ethical relativism
Eugenics
Evolution of societies
Freedom of thought
Freethought
Garden of Eden
Mutation
Natural philosophy
Neo-Creationism
Peter Kropotkin
John Ruskin
Scientific scepticism
Social Darwinism
Social ecology
Social evolutionism
Social implications of the theory of evolution
Universal Darwinism
Notes[edit]
Jump up ^ Though Spencer was an advocate of the inheritance of acquired characters, he considered Lamarck’s failure to explain organic evolution in physical terms as a serious weakness of his theory.[12]
References[edit]
^ Jump up to: a b c d "Letter 5140 – Wallace, A. R. to Darwin, C. R., 2 July 1866". Darwin Correspondence Project. Retrieved 12 January 2010.
"Letter 5145 – Darwin, C. R. to Wallace, A. R., 5 July (1866)". Darwin Correspondence Project. Retrieved 12 January 2010.
^ "Herbert Spencer in his Principles of Biology of 1864, vol. 1, p. 444, wrote: 'This survival of the fittest, which I have here sought to express in mechanical terms, is that which Mr. Darwin has called "natural selection", or the preservation of favoured races in the struggle for life.'" Maurice E. Stucke, Better Competition Advocacy, retrieved 29 August 2007 , citing HERBERT SPENCER, THE PRINCIPLES OF BIOLOGY 444 (Univ. Press of the Pac. 2002.)
^ Jump up to: a b "This preservation, during the battle for life, of varieties which possess any advantage in structure, constitution, or instinct, I have called Natural Selection; and Mr. Herbert Spencer has well expressed the same idea by the Survival of the Fittest. The term "natural selection" is in some respects a bad one, as it seems to imply conscious choice; but this will be disregarded after a little familiarity." Darwin, Charles (1868), The Variation of Animals and Plants under Domestication, 1 (1st ed.), London: John Murray, p. 6, retrieved 10 August 2015
^ Jump up to: a b Freeman, R. B. (1977), "On the Origin of Species", The Works of Charles Darwin: An Annotated Bibliographical Handlist (2nd ed.), Cannon House, Folkestone, Kent, England: Wm Dawson & Sons Ltd
^ Jump up to: a b "This preservation of favourable variations, and the destruction of injurious variations, I call Natural Selection, or the Survival of the Fittest." – Darwin, Charles (1869), On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life (5th ed.), London: John Murray, pp. 91–92, retrieved 22 February 2009
^ Jump up to: a b c "Stephen Jay Gould, Darwin's Untimely Burial", 1976; from Philosophy of Biology:An Anthology, Alex Rosenberg, Robert Arp ed., John Wiley & Sons, May 2009, pp. 99–102.
Jump up ^ "Evolutionary biologists customarily employ the metaphor 'survival of the fittest,' which has a precise meaning in the context of mathematical population genetics, as a shorthand expression when describing evolutionary processes." Chew, Matthew K.; Laubichler, Manfred D. (4 July 2003), "PERCEPTIONS OF SCIENCE: Natural Enemies — Metaphor or Misconception?", Science, 301 (5629): 52–53, doi:10.1126/science.1085274, PMID 12846231, retrieved 20 March 2008
Jump up ^ Vol. 1, p. 444
Jump up ^ U. Kutschera (14 March 2003), A Comparative Analysis of the Darwin-Wallace Papers and the Development of the Concept of Natural Selection (PDF), Institut für Biologie, Universität Kassel, Germany, archived from the original (PDF) on 14 April 2008, retrieved 20 March 2008
Jump up ^ Darwin, Charles (1869), On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life (5th ed.), London: John Murray, p. 72
Jump up ^ The principle of natural selection applied to groups of individual is known as Group selection.
Jump up ^ Herbert Spencer; Truxton Beale (1916), The Man Versus the State: A Collection of Essays, M. Kennerley (snippet)
Jump up ^ Federico Morganti (May 26, 2013). "Adaptation and Progress: Spencer's Criticism of Lamarck". Evolution & Cognition. External link in |publisher= (help)
Jump up ^ Colby, Chris (1996–1997), Introduction to Evolutionary Biology, TalkOrigins Archive, retrieved 22 February 2009
^ Jump up to: a b von Sydow, M. (2014). ‘Survival of the Fittest’ in Darwinian Metaphysics – Tautology or Testable Theory? (pp. 199–222) In E. Voigts, B. Schaff & M. Pietrzak-Franger (Eds.). Reflecting on Darwin. Farnham, London: Ashgate.
^ Jump up to: a b Sahney, S., Benton, M.J. and Ferry, P.A. (2010), "Links between global taxonomic diversity, ecological diversity and the expansion of vertebrates on land" (PDF), Biology Letters, 6 (4): 544–547, doi:10.1098/rsbl.2009.1024, PMC 2936204  , PMID 20106856. CS1 maint: Multiple names: authors list (link)
^ Jump up to: a b John S. Wilkins (1997), Evolution and Philosophy: Social Darwinism – Does evolution make might right?, TalkOrigins Archive, retrieved 21 November 2007
Jump up ^ Leonard, Thomas C. (2005), "Mistaking Eugenics for Social Darwinism: Why Eugenics is Missing from the History of American Economics" (PDF), History of Political Economy, 37 (supplement:): 200–233, doi:10.1215/00182702-37-Suppl_1-200
Jump up ^ Alan Keyes (7 July 2001), WorldNetDaily: Survival of the fittest?, WorldNetDaily, retrieved 19 November 2007
Jump up ^ Mark Isaak (2004), CA002: Survival of the fittest implies might makes right, TalkOrigins Archive, retrieved 19 November 2007
^ Jump up to: a b c d Corey, Michael Anthony (1994), "Chapter 5. Natural Selection", Back to Darwin: the scientific case for Deistic evolution, Rowman and Littlefield, p. 147, ISBN 978-0-8191-9307-0
Jump up ^ Cf. von Sydow, M. (2012). From Darwinian Metaphysics towards Understanding the Evolution of Evolutionary Mechanisms. A Historical and Philosophical Analysis of Gene-Darwinism and Universal Darwinism. Universitätsverlag Göttingen.
Jump up ^ Shermer, Michael; Why People Believe Weird Things; 1997; Pages 143–144
External links[edit]
Origins of the phrase[edit]
AboutDarwin.com — Darwin's Timeline
Pioneers of Psychology
Evolution Quotations compiled by GIGA
Tautology links[edit]
Darwin's Untimely Burial by Stephen Jay Gould
Evolution and Philosophy: A Good Tautology is Hard to Find by John Wilkins, part of the talk.origins archive.
CA500: "Survival of the fittest is a tautology" from the talk.origins index to creationist claims by Mark Ridley.
Is "survival of the fittest" a tautology by Don Lindsay.
Darwin's Great Tautology by the Doubting Thomas
Morality link[edit]
CA002: Survival of the fittest implies that "might makes right"
David Hume — Dialogues Concerning Natural Religion
Evolution and philosophy — Does evolution make might right? by John S. Wilkins.
"Survival of the fittest" by Alan Keyes.
Darwinism and Laissez-Faire Capitalism from the Institute for Creation Research
Kropotkin: Mutual Aid[edit]
Mutual Aid; a factor of evolution at Project Gutenberg
Mutual Aid: A Factor of Evolution – HTML version at the Anarchy Archives
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text:
List of Japanese deities
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This is a list of divinities native to Japanese beliefs and religious traditions. Many of these are from Shinto, while others were imported via Buddhism or Taoism and "integrated" into Japanese mythology and folklore.
Contents
[hide]
1 Major kami
2 Minor kami
3 Buddhism
4 Seven Lucky Gods
5 See also
6 References
Major kami[edit]
Amaterasu-Ō-Mi-Kami (天照大神 or 天照大御神) Commonly called Amaterasu, she is the goddess of the sun as well as the purported ancestress of the Imperial Household of Japan. Her full name means "Great Goddess" or "Great Spirit Who Shines in the Heavens"; she may also be referred to as Ōhiru-menomuchi-no-kami (大日孁貴神). For many reasons, one among them being her ties to the Imperial family, she is often considered (though not officially) to be the "primary god" of Shinto.
Ame-no-Uzume (天宇受売命 or 天鈿女命) Commonly called Uzume, she is the goddess of dawn and revelry, instrumental to the "missing sun motif" in Shinto.[citation needed] She is also known as The Great Persuader and The Heavenly Alarming Female.
Fūjin (風神) Also known as Kami-no-Kaze, he is the Japanese god of the wind and one of the eldest Shinto gods, said to have been present at the creation of the world. He is often depicted as an oni with a bag slung over his back.
Hachiman (八幡神) Also known as Hachiman-shin or Yawata no Kami, he is the god of war and the divine protector of Japan and its people. Originally an agricultural deity, he later became the guardian of the Minamoto clan. His symbolic animal and messenger is the dove.
Inari Ōkami (稲荷大神) The god or goddess of rice and fertility. Their messengers and symbolic animal are foxes. They are often identified with the Ukanomitama and Buddhist deity Dakiniten.
Izanagi (伊弊諾 or 伊邪那岐) The forefather of the gods, he is the first male as well as the god of creation and life. He and his wife, Izanami, were responsible for the birth of the islands of Japan and many kami, though she died in childbirth. Later, after his failed attempt to retrieve her from the underworld, he sired Amaterasu, Susanoo and Tsukuyomi.
Izanami (伊弉冉 or 伊邪那美) Izanagi's wife and sister, she is the first female as well as the goddess of creation and death. She died shortly after the birth of Kagu-tsuchi, and Izanagi followed her to the underworld, but failed to bring her back to the living world. A marital spat between the pair caused the cycle of life and death for all living beings.
Kuninotokotachi (国之常立神?, Kuninotokotachi-no-Kami, in Kojiki)(国常立尊?, Kuninotokotachi-no-Mikoto, in Nihonshoki) is one of the two gods born from "something like a reed that arose from the soil" when the earth was chaotic. In the Nihon Shoki, he is the first of the first three divinities born after heaven and earth were born out of chaos, and is born from something looking like a reed-shoot growing between heaven and earth. He is known by mythology to reside on top of Mount Fuji (富士山). Kuninotokotachi is described as a hitorigami and genderless in Kojiki, while as a male god in Nihon Shoki. Yoshida Kanetomo, the founder of the Yoshida Shintō sect, identified Kuninotokotachi with Amenominakanushi and regarded him as the primordial god of the Universe.
Ninigi-no-Mikoto (瓊瓊杵尊) Commonly called Ninigi, he was the grandson of Amaterasu. His great-grandson was Kan'yamato Iwarebiko, later to be known as Emperor Jimmu, first emperor of Japan.
Ōkuninushi (大国主) A god of nation-building, farming, business, and medicine.
Omoikane (思兼) The deity of wisdom and intelligence, who is always called upon to "ponder" and give good counsel in the deliberations of the heavenly deities.
Raijin (雷神) Commonly called Raiden (雷電), he is the god of thunder and lightning, and is often paired with Fūjin. As with the latter, Raijin is usually depicted as an oni.
Ryūjin (龍神) Also known as Ōwatatsumi, he is a dragon, as well as god of the sea. He resides in Ryūgū-jō, his palace under the sea built out of red and white coral, from where he controls the tides with magical tide jewels. His great-grandson would become Emperor Jimmu.
Suijin (水神) The God of Water.
Susanoo-no-Mikoto (須佐之男尊) Alternately romanized as Susano-o, Susa-no-o, and Susanowo. Reportedly called "Futsushi". He is the god of storms as well as in some cases the god of the sea. He is also somewhat of a trickster god, as Japanese mythology extensively documents the "sibling rivalry" between him and Amaterasu. Susanoo also was responsible for the slaying of the monster Yamata no Orochi and the subsequent discovery of the sacred sword Kusanagi.
Tenjin (天神) The god of scholarship, he is the deified Sugawara no Michizane (845–c903), who was elevated to his position after dying in exile and subsequent disasters in Heiankyo were attributed to his angered spirit.
Toyotama-hime (豊玉姫) Also known as Otohime (乙姫), she was the daughter of Ryūjin and the grandmother of Jimmu. It is said that after she gave birth to her son, she turned into a dragon and disappeared.
Tsukuyomi-no-Mikoto (月読の命 or 月夜見の尊) Also known as Tsukiyomi, Tsuki no Kami, Tsukiyomino Mikoto, and Tsukiyumi no Mikoto, he is the god of the moon. He killed the goddess of food, Uke Mochi, out of disgust and anger in the way she had prepared a meal. This caused Amaterasu to never face him again, causing the sun and moon to be in different parts of the sky.
Takemikazuchi, known as a god of thunder and the god of swords.
Takeminakata
Minor kami[edit]
Amatsu-Mikaboshi (天津甕星), the kami of stars who existed before the Kotoamatsukami. Mentioned under Takemikazuchi.
Ajisukitakahikone
Ame-no-hohi (天菩比神, 天穂日命)
Ame-no-Koyane (天児屋命 or 天児屋根命) A male deity, he is considered the "First in Charge of Divine Affairs", as well as the aide to the first Emperor of Japan.[1] He is also considered to be the ancestor of the Fujiwara family.
Ame-no-oshihomimi (天忍穂耳命)
Ame-no-wakahiko (天若日子, 天稚彦)
Azumi-no-isora (阿曇磯良) is a kami of the seashore. He is considered to be the ancestor of the Azumi people.
Futsunushi (経津主神) Main deity at Katori Shrine.
Iwai-nushi-no-kami (斎主尊), god name whose identity is obscure. Nihon shoki calls him Iwai-no-ushi (斎之大人) and locates him in Katori which suggest the god might be Futsunushi. But there is reason to believe he might be Takehazuchi (建葉槌命) (See Takemizuchi), and there are others who say this might even be Amaterasu.[2]Futsunushi (経津主神)
Isetsuhiko (伊勢都彦命)
Ishikori-dome no Mikoto (石凝姥命), god of metalworking.
Kagu-tsuchi is the kami of fire, and a patron of blacksmiths and potters.
Kotoshironushi
Kuebiko (久延毘古) god of knowledge and agriculture, represented in Japanese mythology as a scarecrow who cannot walk but has comprehensive awareness.
Kuraokami (闇龗), Okami (龗), or Okami no kami (淤加美神) is a legendary Japanese dragon and Shinto deity of rain and snow.
Kukurihime no Kami (菊理媛神), a goddess enshrined at Shirayama Hime Shrine.
Kōjin (三宝荒神), is the god of fire, the hearth, and the kitchen.
Konohanasakuya-hime (木花之開耶姫), the wife of Ninigi and daughter of Ōyamatsumi, and great-grandmother of Jimmu. She is also known as the goddess of Mount Fuji. She is also known by the name Sengen.
Moreya
Nigihayahi-no-mikoto (饒速日尊, 天照国照彦天火明櫛玉饒速日尊) reportedly called "Furu" and said a son of "Futsushi". Only Nigihayahi (Furu) has the name of "Amateru (天照)" among Japanese deities. It is clearly described that Nigihayahi was ruling Yamato (ancient name of the capital and the center of Japan) before conquest war of emperor Jinmu in the two oldest official history books of Japan, "Kojiki" (712) or "Nihon Shoki" (720). According to Nihon Ōdai Ichiran, Nigihayahi descended from the Heaven and was then served by and under the protection of Nagasunebiko, but after having been conquered, the relics Jingi brought by Nigihayahi as proof of his legitimacy were proven inferior to the mere bow and arrows of Emperor Jimmu. His descendants became thus the servants of the Imperial Dynasty.
Oda Nobunaga enshrined at Kenkun-jinja.
Omoikane (思兼神), god of wisdom and intelligence.
Ōyamatsumi (大山積神), an elder brother of Amaterasu, and an important god who rules mountain, sea, and war, as well as the father of Konohanasakuya-hime.
Sukuna-Biko-Na (少名毘古那) A small deity of medicine and rain, who created and solidified the land with Ōkuninushi.
Sarutahiko Ōkami (猿田毘古神), a kami of the Earth that guided Ninigi to the Japanese islands.
Shōtoku Taishi was sometimes worshipped by Shintoists in Prince's Hall (太子堂 Taishido) as the Kami of building trade and easy birth, like in the Hokai-ji of Kamakura.
Seidai Myōjin, god of sports, enshrined at Shiramine Shrine in Kyoto, especially worshipped for Kemari and Football.
Satimasuma (天苗加命) reportedly called "Futsushi" and said a son or elder brother of "Futsu".
Tajimamori (田道間守), god who obtained the tokijiku no kagu no mi in Tokoyo-no-kuni, and hailed as "god of wagashi" (sweets, confections).
Tōshō Daigongen (Tokugawa Ieyasu) enshrined at Nikkō Tōshō-gū and similar shrines.
Toyotomi Hideyoshi enshrined at Toyokuni-jinja.
Ugajin, a harvest and fertility kami represented with the body of a snake and head of a man or woman. S/he may be derived from Ukanomitama.
Ukanomitama, "the spirit of rice in storehouses", a kami associated with food and agriculture.
Uke Mochi (保食神), sometimes called Ogetsu-hime-no-Kami, a goddess of food. After she had spat a fish, vomited or defecated game and coughed rice, she had been killed by a disgusted Tsukuyomi, or in some other versions, Susanoo.
Yama-no-kami are a group of mountain deities, usually represented as female.
Buddhism[edit]
Further information: Japanese Buddhist pantheon
Aizen Myō-ō (愛染明王), a Wisdom King known to transform earthly desires (love/lust) into spiritual awakening.
Amida Nyorai (無量光佛 or 無量壽佛), commonly referred to as Amida-butsu (阿弥陀如来), he is the primary Buddha of the Pure Land school of Buddhism. He is believed to possess infinite meritorious qualities, and is known as the "Lord of the Beyond and the Afterlife". He is one of the Five Dhyani Buddhas.
Daruma (ダルマ), traditionally held in Buddhist mythology to be the founder of Zen Buddhism, as well as the founder of Shaolin Kung Fu. One legend reports that after years of facing a wall in meditation, Bodhidharma's legs and arms fall off due to atrophy. Daruma dolls were created in honor of this legend.
Fudō Myōō (不動明王), a fierce and wrathful Wisdom King who protects all by burning away impediments and defilements, and aiding them towards enlightenment.
Idaten (韋駄天), guardian of Buddhist monasteries and monks.
Jizō (地蔵), a Bodhisattva known as the protector of the vulnerable, especially children, travelers, and expectant mothers. He is also regarded as the patron deity of deceased children and aborted fetuses, and the savior of hell-beings. His statues are a common sight, especially by roadsides and in graveyards.
Kangiten, god (deva) of bliss.
Kannon (観音), a Bodhisattva associated with compassion. Commonly known in English as the "Goddess of Mercy".
Yakushi Nyorai (薬師如来), a Buddha known for healing and medicine.
Seven Lucky Gods[edit]
Main article: Seven Lucky Gods
The Seven Lucky Gods (七福神, Shichi Fukujin) are:
Benzaiten (弁才天 or 弁財天) Also known as Benten or Benzaitennyo, she is the goddess of everything that flows: words (and knowledge, by extension), speech, eloquence, and music. Said to be the third daughter of the dragon-king of Munetsuchi, over the course of years she has gone from being a protective deity of Japan to one who bestows good fortune upon the state and its people. Derived from Saraswati, the equivalent Hindu goddess.[citation needed]
Bishamonten (毘沙門天) Also called Bishamon or Tamonten, he is the god of fortunate warriors and guards, as well as the punisher of criminals. Said to live halfway down the side of Mount Sumeru, the small pagoda he carries symbolizes the divine treasure house that he both guards and gives away its contents.
Daikokuten (大黒天) Often shortened to simply Daikoku, he is variously considered to be the god of wealth (more specifically, the harvest), or of the household (particularly the kitchen). He is recognised by his wide face, smile, and flat black hat. He is often portrayed holding a golden mallet, seated on bales of rice, with mice nearby (which signify plentiful food).
Ebisu (恵比須, 恵比寿, 夷 or 戎) The sole member of the gods believed to have originated in Japan, he was originally known as Hiruko (蛭子), the first child of Izanagi and Izanami. Said to be born without bones, he eventually overcame his handicaps to become the mirthful and auspicious Ebisu (hence one of his titles, "The Laughing God"). He is often depicted holding a rod and a large red sea bream or sea bass. Jellyfish are also associated with this god and the fugu restaurants of Japan will often incorporate Yebisu in their motif.
Fukurokuju (福禄寿) Often confused with Jurōjin, he is the god of wisdom and longevity and said to be an incarnation of the Southern Polestar. He is a star god accompanied by a crane and a turtle, which are considered to be symbols of longevity, and also sometimes accompanied by a black deer. The sacred book tied to his staff is said to contain the lifespan of every person on Earth.
Hotei (布袋) Best known in the Western world as the Laughing Buddha, Hotei is likely the most popular of the gods. His image graces many temples, restaurants, and amulets. Originally based on a Chinese Chan monk, Hotei has become a deity of contentment and abundance.
Jurōjin (寿老人) Also known as Gama, he represents longevity. He is often seen with a fan and a stave, and accompanied by a black deer.
The goddess Kichijōten (吉祥天), also known as Kisshoutennyo, is sometimes considered to be one of the seven gods,[3] replacing either Jurōjin or Fukurokuju.[4] She embodies happiness, fertility, and beauty. Daikoku sometimes manifests as a female known as Daikokunyo (大黒女) or Daikokutennyo (大黒天女).[5] When Kisshoutennyo is counted among the seven Fukujin[4] and Daikoku is regarded in feminine form,[5] all three of the Hindu Tridevi goddesses are represented in the Fukujin.
See also[edit]
Shinto portal
List of legendary creatures from Japan
References[edit]
Jump up ^ http://eos.kokugakuin.ac.jp/modules/xwords/entry.php?entryID=37
Jump up ^ 平林, 章仁 (Akihito Hirabayashi) (1974). "『日本書紀』＜大人＞の用字について". 古代文化. 古代學協會. 26 (3): 161–. catalog "神を斎主する神を斎の大人とよんだと伝えるのであるが、斎主される神とは具体的神名は伝えられていないが、おそらく天神(天照大神)であったろう"
Jump up ^ "Kisshōten (Kichijōten)". Retrieved 14 December 2015.
^ Jump up to: a b "Butsuzōzui (Illustrated Compendium of Buddhist Images)" (digital photos) (in Japanese). Ehime University Library. 1796. p. (077.jpg).
^ Jump up to: a b "Butsuzōzui (Illustrated Compendium of Buddhist Images)" (digital photos) (in Japanese). Ehime University Library. 1796. p. (059.jpg).
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Hermitian matrix
From Wikipedia, the free encyclopedia
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For matrices with symmetry over the real number field, see symmetric matrix.
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:
a i j = a j i ¯ {\displaystyle a_{ij}={\overline {a_{ji}}}} or A = A T ¯ {\displaystyle A={\overline {A^{\text{T}}}}} , in matrix form.
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H {\displaystyle A^{\text{H}}} , then the Hermitian property can be written concisely as
A = A H . {\displaystyle A=A^{\text{H}}.}
Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A H = A † = A ∗ {\displaystyle A^{\text{H}}=A^{\dagger }=A^{\ast }} , although note that in quantum mechanics, A ∗ {\displaystyle A^{\ast }} typically means the complex conjugate only, and not the conjugate transpose.
Contents
1 Applications
2 Examples
3 Properties
4 Further properties
5 Rayleigh quotient
6 See also
7 References
8 External links
Applications[edit]
Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.
Examples[edit]
In this section, the conjugate transpose of matrix A {\displaystyle A} is denoted as A H {\displaystyle A^{\text{H}}} , the transpose of matrix A {\displaystyle A} is denoted as A T {\displaystyle A^{\text{T}}} and conjugate of matrix A {\displaystyle A} is denoted as A ¯ {\displaystyle {\overline {A}}} .
See the following example:
[ 2 2 + i 4 2 − i 3 i 4 − i 1 ] {\displaystyle {\begin{bmatrix}2&2+i&4\\2-i&3&i\\4&-i&1\\\end{bmatrix}}}
The diagonal elements must be real, as they must be their own complex conjugate.
Well-known families of Pauli matrices, Gell-Mann matrices and their generalizations are Hermitian. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices (see below).
Here we offer another useful Hermitian matrix using an abstract example. If a square matrix A {\displaystyle A} equals the multiplication of a matrix and its conjugate transpose, that is, A = B B H {\displaystyle A=BB^{\text{H}}} , then A {\displaystyle A} is a Hermitian positive semi-definite matrix. Furthermore, if B {\displaystyle B} is row full-rank, then A {\displaystyle A} is positive definite.
Properties[edit]
This section needs expansion with: Proof of the properties requested. You can help by adding to it. (February 2018)
The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real.
Proof: By definition of the Hermitian matrix
H i j = H j i ¯ {\displaystyle H_{ij}={\overline {H_{ji}}}}
so for i = j the above follows.
Note that only complex-valued entries on the main diagonal are forbidden; Hermetian matricies can have complex-valued entries in their off-diagonal elements, as long as diagonally-opposite entries are complex conjugates.
A matrix that has only real entries is Hermitian if and only if it is symmetric. A real and symmetric matrix is simply a special case of a Hermitian matrix.
Proof: H i j = H j i ¯ {\displaystyle H_{ij}={\overline {H_{ji}}}} by definition. Thus Hij = Hji (matrix symmetry) if and only if H i j = H i j ¯ {\displaystyle H_{ij}={\overline {H_{ij}}}} (Hij is real).
Every Hermitian matrix is a normal matrix. That is to say, AAH = AHA.
Proof: A = AH, so AAH = AA = AHA.
The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of ℂn consisting of n eigenvectors of A.
The sum of any two Hermitian matrices is Hermitian.
Proof: ( A + B ) i j = A i j + B i j = A j i ¯ + B j i ¯ = ( A + B ) j i ¯ {\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A_{ji}}}+{\overline {B_{ji}}}={\overline {(A+B)_{ji}}}} , as claimed.
The inverse of an invertible Hermitian matrix is Hermitian as well.
The product of two Hermitian matrices A and B is Hermitian if and only if AB = BA.
Proof: Note that ( A B ) H = ( A B ) T ¯ = B T A T ¯ = B T ¯ A T ¯ = B H A H = B A . {\displaystyle (AB)^{H}={\overline {(AB)^{T}}}={\overline {B^{T}A^{T}}}={\overline {B^{T}}}{\overline {A^{T}}}=B^{H}A^{H}=BA.} Thus ( A B ) H = A B {\displaystyle (AB)^{H}=AB} if and only if A B = B A {\displaystyle AB=BA} .
Thus An is Hermitian if A is Hermitian and n is an integer.
For an arbitrary complex valued vector v the product v H A v {\displaystyle v^{\text{H}}Av} is real because of v H A v = ( v H A v ) H {\displaystyle v^{\text{H}}Av=(v^{\text{H}}Av)^{\text{H}}} . This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system e.g. total spin which have to be real.
The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, ℂ, since the identity matrix In is Hermitian, but i In is not. However the complex Hermitian matrices do form a vector space over the real numbers ℝ. In the 2n2-dimensional vector space of complex n × n matrices over ℝ, the complex Hermitian matrices form a subspace of dimension n2. If Ejk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis (orthonormal w.r.t. the Frobenius inner product) can be described as follows:
E j j {\displaystyle \;E_{jj}} for 1 ≤ j ≤ n {\displaystyle 1\leq j\leq n} (n matrices)
together with the set of matrices of the form
1 2 ( E j k + E k j ) {\displaystyle \;{\frac {1}{\sqrt {2}}}(E_{jk}+E_{kj})} for 1 ≤ j < k ≤ n {\displaystyle 1\leq j<k\leq n} (n2 − n/2 matrices)
and the matrices
i 2 ( ( E j k − E k j ) {\displaystyle \;{\frac {i}{\sqrt {2}}}((E_{jk}-E_{kj})} for 1 ≤ j < k ≤ n {\displaystyle 1\leq j<k\leq n} (n2 − n/2 matrices)
where i {\displaystyle i} denotes the complex number − 1 {\displaystyle {\sqrt {-1}}} , called the “imaginary unit”.
If n orthonormal eigenvectors u 1 , … , u n {\displaystyle u_{1},\dots ,u_{n}} of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is A = U Λ U H {\displaystyle A=U\Lambda U^{\text{H}}} where U U H = I = U H U {\displaystyle UU^{\text{H}}=I=U^{\text{H}}U} and therefore
A = ∑ j λ j u j u j H {\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\text{H}}} ,
where λ j {\displaystyle \lambda _{j}} are the eigenvalues on the diagonal of the diagonal matrix Λ {\displaystyle \;\Lambda } .
Further properties[edit]
Additional facts related to Hermitian matrices include:
The sum of a square matrix and its conjugate transpose ( A + A H ) {\displaystyle (A+A^{\text{H}})} is Hermitian.
The difference of a square matrix and its conjugate transpose ( A − A H ) {\displaystyle (A-A^{\text{H}})} is skew-Hermitian (also called antihermitian). This implies that commutator of two Hermitian matrices is skew-Hermitian.
An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:
C = A + B with A = 1 2 ( C + C H ) and B = 1 2 ( C − C H ) {\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}(C+C^{\text{H}})\quad {\mbox{and}}\quad B={\frac {1}{2}}(C-C^{\text{H}})}
The determinant of a Hermitian matrix is real:
Proof: det ( A ) = det ( A T ) ⇒ det ( A H ) = det ( A ) ¯ {\displaystyle \det(A)=\det(A^{\mathrm {T} })\quad \Rightarrow \quad \det(A^{\text{H}})={\overline {\det(A)}}}
Therefore if A = A H ⇒ det ( A ) = det ( A ) ¯ {\displaystyle A=A^{\text{H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}} .
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
Rayleigh quotient[edit]
Main article: Rayleigh quotient
In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[3] R ( M , x ) {\displaystyle R(M,x)} , is defined as:[4][5]
R ( M , x ) := x H M x x H x {\displaystyle R(M,x):={x^{\text{H}}Mx \over x^{\text{H}}x}} .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x H {\displaystyle x^{\text{H}}} to the usual transpose x T {\displaystyle x^{\text{T}}} . Note that R ( M , c x ) = R ( M , x ) {\displaystyle R(M,cx)=R(M,x)} for any non-zero real scalar c {\displaystyle c} . Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M) when x {\displaystyle x} is v min {\displaystyle v_{\min }} (the corresponding eigenvector). Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,x)\leq \lambda _{\max }} and R ( M , v max ) = λ max {\displaystyle R(M,v_{\max })=\lambda _{\max }} .
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra.
See also[edit]
Skew-Hermitian matrix (anti-Hermitian matrix)
Haynsworth inertia additivity formula
Hermitian form
Self-adjoint operator
Unitary matrix
References[edit]
Jump up ^ Frankel, Theodore (2004). The Geometry of Physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7.
Jump up ^ Physics 125 Course Notes at California Institute of Technology
Jump up ^ Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh.
Jump up ^ Horn, R. A. and C. A. Johnson. 1985. Matrix Analysis. Cambridge University Press. pp. 176–180.
Jump up ^ Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Hermitian matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
"Hermitian Matrices". MathPages.com.
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