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Self-reference occurs in
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
or
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of s ...
s 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 communications and information processing, code is a system of rules to convert information—such as a letter (alphabet), letter, word, sound, image, or gesture—into another form, sometimes data compression, shortened or secrecy, secret ...
. In philosophy, it 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 Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
,
second-order cybernetics Second-order cybernetics, also known as the cybernetics of cybernetics, is the recursive application of cybernetics to itself and the reflexive practice of cybernetics according to such a critique. It is cybernetics where "the role of the observer ...
, and
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Lingu ...
, as well as in humor. Self-referential statements are sometimes
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
ical, 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 In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consisten ...
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 In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contain ...
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 Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
, 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 Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...
(related to the mathematical
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
) in
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that ...
, 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 Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, 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 In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input. Etymology Tall boots may have a tab, loop or handle at the top known as a bootstrap, allowing one to use fingers ...
.
Self-modifying code In computer science, self-modifying code (SMC) is code that alters its own instructions while it is executing – usually to reduce the instruction path length and improve performance or simply to reduce otherwise repetitively similar code ...
is possible to write (programs which operate on themselves), both with assembler and with functional languages such as
Lisp A lisp is a speech impairment in which a person misarticulates sibilants (, , , , , , , ). These misarticulations often result in unclear speech. Types * A frontal lisp occurs when the tongue is placed anterior to the target. Interdental lispi ...
, 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 A pine is any conifer tree or shrub in the genus ''Pinus'' () of the family (biology), family Pinaceae. ''Pinus'' is the sole genus in the subfamily Pinoideae. The World Flora Online created by the Royal Botanic Gardens, Kew and Missouri Botanic ...
('Pine is not Elm'). The
GNU Hurd GNU Hurd is a collection of microkernel servers written as part of GNU, for the GNU Mach microkernel. It has been under development since 1990 by the GNU Project of the Free Software Foundation, designed as a replacement for the Unix kernel, ...
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 biology

The biology of
self-replication Self-replication is any behavior of a dynamical system that yields construction of an identical or similar copy of itself. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and c ...
is self-referential, as embodied by DNA and
RNA Ribonucleic acid (RNA) is a polymeric molecule essential in various biological roles in coding, decoding, regulation and expression of genes. RNA and deoxyribonucleic acid ( DNA) are nucleic acids. Along with lipids, proteins, and carbohydra ...
replication mechanisms. Models of self-replication are found in
Conway's Game of Life The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is a zero-player game, meaning that its evolution is determined by its initial state, requiring no furt ...
and have inspired engineering systems such as the self-replicating 3D printer RepRap.


In art

Self-reference occurs in
literature Literature is any collection of written work, but it is also used more narrowly for writings specifically considered to be an art form, especially prose fiction, drama, and poetry. In recent centuries, the definition has expanded to inclu ...
and film when an author refers to his or her own work in the context of the work itself. Examples include
Miguel de Cervantes Miguel de Cervantes Saavedra (; 29 September 1547 (assumed) – 22 April 1616 NS) was an Early Modern Spanish writer widely regarded as the greatest writer in the Spanish language and one of the world's pre-eminent novelists. He is best know ...
' ''
Don Quixote is a Spanish epic novel by Miguel de Cervantes. Originally published in two parts, in 1605 and 1615, its full title is ''The Ingenious Gentleman Don Quixote of La Mancha'' or, in Spanish, (changing in Part 2 to ). A founding work of Wester ...
'',
Shakespeare William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's natio ...
's ''
A Midsummer Night's Dream ''A Midsummer Night's Dream'' is a comedy written by William Shakespeare 1595 or 1596. The play is set in Athens, and consists of several subplots that revolve around the marriage of Theseus and Hippolyta. One subplot involves a conflict ...
'', '' The Tempest'' and ''
Twelfth Night ''Twelfth Night'', or ''What You Will'' is a romantic comedy by William Shakespeare, believed to have been written around 1601–1602 as a Twelfth Night's entertainment for the close of the Christmas season. The play centres on the twins V ...
'',
Denis Diderot Denis Diderot (; ; 5 October 171331 July 1784) was a French philosopher, art critic, and writer, best known for serving as co-founder, chief editor, and contributor to the ''Encyclopédie'' along with Jean le Rond d'Alembert. He was a promine ...
's '' Jacques le fataliste et son maître'',
Italo Calvino Italo Calvino (, also , ;. RAI (circa 1970), retrieved 25 October 2012. 15 October 1923 – 19 September 1985) was an Italian writer and journalist. His best known works include the '' Our Ancestors'' trilogy (1952–1959), the ''Cosmicomi ...
's '' If on a winter's night a traveler'', many stories by
Nikolai Gogol Nikolai Vasilyevich Gogol; uk, link=no, Мико́ла Васи́льович Го́голь, translit=Mykola Vasyliovych Hohol; (russian: Яновский; uk, Яновський, translit=Yanovskyi) ( – ) was a Russian novelist, ...
, '' Lost in the Funhouse'' by John Barth, Luigi Pirandello's '' Six Characters in Search of an Author'', Federico Fellini's '' '' and Bryan Forbes's '' The L-Shaped Room''. Speculative fiction writer Samuel R. Delany makes use of this in his novels ''
Nova A nova (plural novae or novas) is a transient astronomical event that causes the sudden appearance of a bright, apparently "new" star (hence the name "nova", which is Latin for "new") that slowly fades over weeks or months. Causes of the dramati ...
'' 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 A nova (plural novae or novas) is a transient astronomical event that causes the sudden appearance of a bright, apparently "new" star (hence the name "nova", which is Latin for "new") that slowly fades over weeks or months. Causes of the dramati ...
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 ''Spaceballs'' is a 1987 American space opera parody film co-written, produced and directed by Mel Brooks. It is primarily a parody of the original ''Star Wars'' trilogy, but also parodies other sci-fi films and popular franchises including ...
, Director
Mel Brooks Mel Brooks (born Melvin James Kaminsky; June 28, 1926) is an American actor, comedian and filmmaker. With a career spanning over seven decades, he is known as a writer and director of a variety of successful broad farces and parodies. He began ...
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 Homer (; grc, Ὅμηρος , ''Hómēros'') (born ) was a Greek poet who is credited as the author of the '' Iliad'' and the '' Odyssey'', two epic poems that are foundational works of ancient Greek literature. Homer is considered one of ...
's ''
Iliad The ''Iliad'' (; grc, Ἰλιάς, Iliás, ; "a poem about Ilium") is one of two major ancient Greek epic poems attributed to Homer. It is one of the oldest extant works of literature still widely read by modern audiences. As with the '' Odys ...
'', where
Helen of Troy Helen of Troy, Helen, Helena, (Ancient Greek: Ἑλένη ''Helénē'', ) also known as beautiful Helen, Helen of Argos, or Helen of Sparta, was a figure in Greek mythology said to have been the most beautiful woman in the world. She was believe ...
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 Jorge Francisco Isidoro Luis Borges Acevedo (; ; 24 August 1899 – 14 June 1986) was an Argentine short-story writer, essayist, poet and translator, as well as a key figure in Spanish-language and international literature. His best-known b ...
play with self-reference and related paradoxes in many ways.
Samuel Beckett Samuel Barclay Beckett (; 13 April 1906 – 22 December 1989) was an Irish novelist, dramatist, short story writer, theatre director, poet, and literary translator. His literary and theatrical work features bleak, impersonal and tragicomic ex ...
'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 A creation myth (or cosmogonic myth) is a symbolic narrative of how the world began and how people first came to inhabit it., "Creation myths are symbolic stories describing how the universe and its inhabitants came to be. Creation myths develop ...
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 The ouroboros or uroboros () is an ancient symbol depicting a serpent or dragon eating its own tail. The ouroboros entered Western tradition via ancient Egyptian iconography and the Greek magical tradition. It was adopted as a symbol in Gno ...
is a mythical dragon which eats itself. The
Quran The Quran (, ; Standard Arabic: , Quranic Arabic: , , 'the recitation'), also romanized Qur'an or Koran, is the central religious text of Islam, believed by Muslims to be a revelation from God. It is organized in 114 chapters (pl.: , sing.: ...
includes numerous instances of self-referentiality. The
surrealist Surrealism is a cultural movement that developed in Europe in the aftermath of World War I in which artists depicted unnerving, illogical scenes and developed techniques to allow the unconscious mind to express itself. Its aim was, according to ...
painter
René Magritte René François Ghislain Magritte (; 21 November 1898 – 15 August 1967) was a Belgian surrealist artist known for his depictions of familiar objects in unfamiliar, unexpected contexts, which often provoked questions about the nature and bounda ...
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 A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
. Such sentences can lead to problems, for example, in law, where statements bringing laws into existence can contradict one another or themselves.
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
claimed to have found such a paradox in the
US constitution The Constitution of the United States is the supreme law of the United States of America. It superseded the Articles of Confederation, the nation's first constitution, in 1789. Originally comprising seven articles, it delineates the nation ...
at his citizenship ceremony. Self-reference occasionally occurs in the
media Media may refer to: Communication * Media (communication), tools used to deliver information or data ** Advertising media, various media, content, buying and placement for advertising ** Broadcast media, communications delivered over mass el ...
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 Wikipedia is a multilingual free online encyclopedia written and maintained by a community of volunteers, known as Wikipedians, through open collaboration and using a wiki-based editing system. Wikipedia is the largest and most-read ref ...
'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.


In popular culture

*
Douglas Hofstadter Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, an ...
's books, especially ''
Metamagical Themas ''Metamagical Themas'' is an eclectic collection of articles that Douglas Hofstadter wrote for the popular science magazine ''Scientific American'' during the early 1980s. The anthology was published in 1985 by Basic Books. The volume is subst ...
'' 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
wiki A wiki ( ) is an online hypertext publication collaboratively edited and managed by its own audience, using a web browser. A typical wiki contains multiple pages for the subjects or scope of the project, and could be either open to the pu ...
s and other technologies. Hofstadter's ' strange loop'
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
attempts to map
consciousness Consciousness, at its simplest, is sentience and awareness of internal and external existence. However, the lack of definitions has led to millennia of analyses, explanations and debates by philosophers, theologians, linguisticians, and scien ...
onto self-reference, but is a minority position in
philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the body. The mind–body problem is a paradigmatic issue in philosophy of mind, although a number of other issues are ad ...
. * The subgenre of " recursive
science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imagination, imaginative and futuristic concepts such as advanced science and technology, space exploration, time travel, Paral ...
" 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 are self-referential, making it more difficult to amend themselves than a simple law.


See also


References


Sources

* Bartlett, Steven J. ames(Ed.) (1992). ''Reflexivity: A Source-book in Self-reference''. Amsterdam, North-Holland
(PDF). RePub, Erasmus University
* Hofstadter, D. R. (1980). '' Gödel, Escher, Bach: an Eternal Golden Braid''. New York,
Vintage Books Vintage Books is a trade paperback publishing imprint of Penguin Random House originally established by Alfred A. Knopf in 1954. The company was purchased by Random House in April 1960, and a British division was set up in 1990. After Random Ho ...
. *
Smullyan, Raymond Raymond Merrill Smullyan (; May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher. Born in Far Rockaway, New York, his first career was stage magic. He earned a BSc fro ...
(1994), ''Diagonalization and Self-Reference'', Oxford Science Publications, * Crabtree, Jonathan J. (2016), ''The Lost Logic of Elementary Mathematics and the Haberdasher who Kidnapped Kaizen'', Proceedings of the Mathematical Association of Victoria (MAV) Annual Conference, 53, 98–106, {{Authority control Grammar Theory of computation Logic Philosophy of language