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69-413: A tangled hierarchy is a hierarchical consciousness system in which a strange loop appears. A strange loop is a hierarchy of levels, each of which is linked to at least one other by some type of relationship. A strange loop hierarchy is "tangled" (Hofstadter refers to this as a " heterarchy "), in that there is no well defined highest or lowest level; moving through the levels, one eventually returns to

138-505: A 3D model of a human for example, the chest is a parent of the upper left arm, which is a parent of the lower left arm, which is a parent of the hand . This pattern is used in modeling and animation for almost everything built as a 3D digital model. Many grammatical theories, such as phrase-structure grammar , involve hierarchy. Direct–inverse languages such as Cree and Mapudungun distinguish subject and object on verbs not by different subject and object markers, but via

207-514: A family tree ) and inheritance ( succession and heirship ). All the requisites of a well-rounded life and lifestyle can be organized using Maslow's hierarchy of human needs - according to Maslow's hierarchy of human needs. Learning steps often follow a hierarchical scheme—to master differential equations one must first learn calculus ; to learn calculus one must first learn elementary algebra ; and so on. Nature offers hierarchical structures, as numerous schemes such as Linnaean taxonomy ,

276-475: A reverse hierarchy , the conceptual pyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This mode represents the idea that members of the higher rankings are responsible for the members of the lower rankings. Empirically, when we observe in nature a large proportion of the (complex) biological systems, they exhibit hierarchic structure. On theoretical grounds we could expect complex systems to be hierarchies in

345-442: A closed cycle. That is, despite one's sense of departing ever further from one's origin, one winds up, to one's shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop . (pp. 101–102) According to Hofstadter, strange loops take form in human consciousness as the complexity of active symbols in the brain inevitably leads to the same kind of self-reference which Gödel proved

414-446: A containment hierarchy is demonstrated in class inheritance in object-oriented programming . Two types of containment hierarchies are the subsumptive containment hierarchy and the compositional containment hierarchy. A subsumptive hierarchy " subsumes " its children, and a compositional hierarchy is " composed " of its children. A hierarchy can also be both subsumptive and compositional . A subsumptive containment hierarchy

483-460: A directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: linear and branching . In a linear hierarchy , the maximum degree is 1. In other words, all of the objects can be visualized in a line-up, and each object (excluding the top and bottom ones) has exactly one direct subordinate and one direct superior. This

552-402: A file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder. They will keep opening files within the folders until the designated file is located. For more complicated hierarchies, the stair structure represents hierarchical relationships through the use of visual stacking. Visually imagine the top of a downward staircase beginning at

621-428: A generalized nested hierarchy allows for multiple objects within levels but with each object having only one parent at each level. The general concept is both demonstrated and mathematically formulated in the following example: A square can always also be referred to as a quadrilateral, polygon or shape. In this way, it is a hierarchy. However, consider the set of polygons using this classification. A square can only be

690-511: A hierarchy of persons. In this system, the three (or four with Algonquian languages ) persons occur in a hierarchy of salience . To distinguish which is subject and which object, inverse markers are used if the object outranks the subject. On the other hand, languages include a variety of phenomena that are not hierarchical. For example, the relationship between a pronoun and a prior noun-phrase to which it refers commonly crosses grammatical boundaries in non-hierarchical ways. The structure of

759-444: A high priest', from hierarkhes , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as architecture , philosophy , design , mathematics , computer science , organizational theory , systems theory , systematic biology , and

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828-406: A higher-level entity can have on x' s properties and interactions. Furthermore, the entities found at each level are autonomous . Kulish (2002) suggests that almost every system of organization which humans apply to the world is arranged hierarchically. Some conventional definitions of the terms "nation" and "government" suggest that every nation has a government and that every government

897-554: A lack of pride and low self-esteem can result. In a positive context, self-consciousness may affect the development of identity, for it is during periods of high self-consciousness that people come the closest to knowing themselves objectively. Self-consciousness affects people in varying degrees, as some people are constantly self-monitoring or self-involved, while others are completely oblivious about themselves. Psychologists frequently distinguish between two kinds of self-consciousness, private and public. Private self-consciousness

966-419: A less concrete, more elusive notion. What I mean by "strange loop" is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in an hierarchy, and yet somehow the successive "upward" shifts turn out to give rise to

1035-524: A musical composition is often understood hierarchically (for example by Heinrich Schenker (1768–1835, see Schenkerian analysis ), and in the (1985) Generative Theory of Tonal Music , by composer Fred Lerdahl and linguist Ray Jackendoff ). The sum of all notes in a piece is understood to be an all-inclusive surface, which can be reduced to successively more sparse and more fundamental types of motion. The levels of structure that operate in Schenker's theory are

1104-410: A problem. The use of parentheses is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example: (2 + 5) × (7 - 4). In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with

1173-439: A quadrilateral; it can never be a triangle , hexagon , etc. Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original Linnaean taxonomy (the version he laid out in the 10th edition of Systema Naturae ), a human can be formulated as: Taxonomies may change frequently (as seen in biological taxonomy ), but the underlying concept of nested hierarchies

1242-404: A required general property. These level hierarchies are characterized by bi-directional causation . Upward causation involves lower-level entities causing some property of a higher level entity; children entities may interact to yield parent entities, and parents are composed at least partly by their children. Downward causation refers to the effect that the incorporation of entity x into

1311-477: A series of majority preferences may return to the original candidate, leaving no clear preference by the group. In this case, some candidate beats an opponent, who in turn beats another opponent, and so forth, until a candidate is reached who beats the original candidate. The liar paradox and Russell's paradox also involve strange loops, as does René Magritte 's painting The Treachery of Images . The mathematical phenomenon of polysemy has been observed to be

1380-400: A similar "flipping around of causality" appears to happen in minds possessing self-consciousness ; the mind perceives itself as the cause of certain feelings. The parallels between downward causality in formal systems and downward causality in brains are explored by Theodor Nenu in 2022, together with other aspects of Hofstadter's metaphysics of mind. Nenu also questions the correctness of

1449-580: A similar way a sound with seemingly ever increasing tempo can be constructed, as was demonstrated by Jean-Claude Risset . Visual illusions depicting strange loops include the Penrose stairs and the Barberpole illusion . A quine in software programming is a program that produces a new version of itself without any input from the outside. A similar concept is metamorphic code . Efron's dice are four dice that are intransitive under gambler's preference. I.e.,

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1518-399: A strange loop. At the denotational level, the term refers to situations where a single entity can be seen to mean more than one mathematical object. See Tanenbaum (1999). The Stonecutter is an old Japanese fairy tale with a story that explains social and natural hierarchies as a strange loop. Hierarchy A hierarchy (from Greek : ἱεραρχία , hierarkhia , 'rule of

1587-466: A world in which complexity had to evolve from simplicity. System hierarchies analysis performed in the 1950s, laid the empirical foundations for a field that would become, from the 1980s, hierarchical ecology . The theoretical foundations are summarized by thermodynamics . When biological systems are modeled as physical systems , in the most general abstraction, they are thermodynamic open systems that exhibit self-organised behavior, and

1656-539: Is a branching hierarchy in which at least one object has two parent objects. For example, a graduate student can have two co-supervisors to whom the student reports directly and equally, and who have the same level of authority within the university hierarchy (i.e., they have the same position or tenure status). Possibly the first use of the English word hierarchy cited by the Oxford English Dictionary

1725-441: Is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and " IS-A hierarchy". The last term describes the relationship between each level—a lower-level object "is a" member of the higher class. The taxonomical structure outlined above is a subsumptive containment hierarchy. Using again the example of Linnaean taxonomy, it can be seen that an object that

1794-404: Is a direct extrapolation of the nested hierarchy concept. All of the ordered sets are still nested, but every set must be " strict "—no two sets can be identical. The shapes example above can be modified to demonstrate this: The notation x ⊊ y {\displaystyle x\subsetneq y\,} means x is a subset of y but is not equal to  y . A general example of

1863-480: Is a hierarchical ordering of nested sets . The concept of nesting is exemplified in Russian matryoshka dolls . Each doll is encompassed by another doll, all the way to the outer doll. The outer doll holds all of the inner dolls, the next outer doll holds all the remaining inner dolls, and so on. Matryoshkas represent a nested hierarchy where each level contains only one object, i.e., there is only one of each size of doll;

1932-503: Is a member of the level Mammalia "is a" member of the level Animalia ; more specifically, a human "is a" primate, a primate "is a" mammal, and so on. A subsumptive hierarchy can also be defined abstractly as a hierarchy of " concepts ". For example, with the Linnaean hierarchy outlined above, an entity name like Animalia is a way to group all the species that fit the conceptualization of an animal. A compositional containment hierarchy

2001-407: Is a tendency to introspect and examine one's inner self and feelings. Public self-consciousness is an awareness of the self as it is viewed by others. This kind of self-consciousness can result in self-monitoring and social anxiety . Both private and public self-consciousness are viewed as personality traits that are relatively stable over time, but they are not correlated. Just because an individual

2070-405: Is always transitive . The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects. Mathematically, in its most general form, a hierarchy is a partially ordered set or poset . The system in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with

2139-420: Is always the same. In many programming taxonomies and syntax models (as well as fractals in mathematics), nested hierarchies, including Russian dolls, are also used to illustrate the properties of self-similarity and recursion . Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic. A containment hierarchy

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2208-507: Is an ordering of the parts that make up a system—the system is "composed" of these parts. Most engineered structures, whether natural or artificial, can be broken down in this manner. The compositional hierarchy that every person encounters at every moment is the hierarchy of life . Every person can be reduced to organ systems , which are composed of organs , which are composed of tissues , which are composed of cells , which are composed of molecules , which are composed of atoms . In fact,

2277-475: Is another illustrative example of a strange loop. Named after Roger Shepard , it is a sound consisting of a superposition of tones separated by octaves . When played with the base pitch of the tone moving upwards or downwards, it is referred to as the Shepard scale . This creates the auditory illusion of a tone that continually ascends or descends in pitch, yet which ultimately seems to get no higher or lower. In

2346-484: Is being watched or observed, the feeling that "everyone is looking" at oneself. Some people are habitually more self-conscious than others. Unpleasant feelings of self-consciousness sometimes become associated with shyness or paranoia . Notable opponents of self-consciousness include Thomas Carlyle . When feeling self-conscious, one becomes aware of even the smallest of one's own actions. Such awareness can impair one's ability to perform complex actions. Adolescence

2415-451: Is believed to be a time of heightened self-consciousness. A person with a chronic tendency toward self-consciousness may be shy or introverted . Unlike self-awareness , which in a philosophical context is being conscious of oneself as an individual, self-consciousness – being excessively conscious of one's appearance or manner – can be a problem at times. Self-consciousness is often associated with shyness and embarrassment, in which case

2484-483: Is credited with first use of it as an abstract noun. Since hierarchical churches, such as the Roman Catholic (see Catholic Church hierarchy ) and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with God as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in secular settings. A hierarchy

2553-448: Is false ," wherein the sentence's basis of truth is found in referring to itself and its assertion, causing a logical paradox. Hofstadter argues that the psychological self arises out of a similar kind of paradox. The brain is not born with an "I" – the ego emerges only gradually as experience shapes the brain's dense web of active symbols into a tapestry rich and complex enough to begin twisting back upon itself . According to this view,

2622-631: Is hierarchical. Sociologists can analyse socioeconomic systems in terms of stratification into a social hierarchy (the social stratification of societies), and all systematic classification schemes ( taxonomies ) are hierarchical. Most organized religions , regardless of their internal governance structures, operate as a hierarchy under deities and priesthoods . Many Christian denominations have an autocephalous ecclesiastical hierarchy of leadership . Families can be viewed as hierarchical structures in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in

2691-598: Is infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point. Organizations can be structured as a dominance hierarchy . In an organizational hierarchy, there is a single person or group with the most power or authority , and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including governments , companies , armed forces , militia and organized religions . The units or persons within an organization may be depicted hierarchically in an organizational chart . In

2760-577: Is not to be confused with consciousness in the sense of qualia . Historically, "self-consciousness" was synonymous with " self-awareness ", referring to a state of awareness that one exists and that one has consciousness . While "self-conscious" and "self-aware" are still sometimes used interchangeably, particularly in philosophy , "self-consciousness" has commonly come to refer to a preoccupation with oneself, especially with how others might perceive one's appearance or one's actions. An unpleasant feeling of self-consciousness may occur when one realizes that one

2829-420: Is one such form. Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed (see below ). In an organizational context, the following terms are often used related to hierarchies: In a mathematical context (in graph theory ), the general terminology used is different. Most hierarchies use a more specific vocabulary pertaining to their subject, but

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2898-590: Is referring to the objects and not the levels ; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects. In a branching hierarchy , one or more objects has a degree of 2 or more (and therefore the minimum degree is 2 or higher). For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy. Branching hierarchies are present within numerous systems, including organizations and classification schemes . The broad category of branching hierarchies can be further subdivided based on

2967-420: Is typically depicted as a pyramid , where the height of a level represents that level's status and width of a level represents the quantity of items at that level relative to the whole. For example, the few Directors of a company could be at the apex , and the base could be thousands of people who have no subordinates. These pyramids are often diagrammed with a triangle diagram which serves to emphasize

3036-484: The "postal district" , consists of 18 objects (letters). The next level down is the "zone", where the objects are the digits 0–9. This is an example of an overlapping hierarchy , because each of these 10 objects has 18 parents. The hierarchy continues downward to generate, in theory, 7,200,000 unique codes of the format A0A 0A0 (the second and third letter positions allow 20 objects each). Most library classification systems are also hierarchical. The Dewey Decimal System

3105-423: The liar paradox as examples that illustrate the idea of strange loops, which is expressed fully in the proof of Gödel 's incompleteness theorem . The " chicken or the egg " paradox is perhaps the best-known strange loop problem. The " ouroboros ", which depicts a dragon eating its own tail, is perhaps one of the most ancient and universal symbolic representations of the reflexive loop concept. A Shepard tone

3174-472: The organization of life , and biomass pyramids attempt to document. While the above examples are often clearly depicted in a hierarchical form and are classic examples, hierarchies exist in numerous systems where this branching structure is not immediately apparent. For example, most postal-code systems are hierarchical. Using the Canadian postal code system as an example, the top level's binding concept,

3243-574: The set/subset relations between dissipative structures can be characterized in a hierarchy. Other hierarchical representations related to biology include ecological pyramids which illustrate energy flow or trophic levels in ecosystems , and taxonomic hierarchies, including the Linnean classification scheme and phylogenetic trees that reflect inferred patterns of evolutionary relationship among living and extinct species. CGI and computer-animation programs mostly use hierarchies for models. On

3312-473: The social sciences (especially political science ). A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates , although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in

3381-634: The above quote by focusing on the sentence which "says about itself" that it is provable (also known as a Henkin-sentence, named after logician Leon Henkin ). It turns out that under suitable meta-mathematical choices (where the Hilbert-Bernays provability conditions do not obtain), one can construct formally undecidable (or even formally refutable) Henkin-sentences for the arithmetical system under investigation. This system might very well be Hofstadter's Typographical Number Theory used in Gödel, Escher, Bach or

3450-420: The brains of others, and likely even in artificial brains . The "strangeness" of a strange loop comes from the brain's perception, because the brain categorizes its input in a small number of "symbols" (by which Hofstadter means groups of neurons standing for something in the outside world). So the difference between the video-feedback loop and the brain's strange loops, is that while the former converts light to

3519-515: The degree. A flat hierarchy (also known for companies as flat organization ) is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span. Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example, diamonds and graphite are flat hierarchies of numerous carbon atoms that can be further decomposed into subatomic particles. An overlapping hierarchy

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3588-462: The dice are ordered A > B > C > D > A , where x > y means "a gambler prefers x to y ". Individual preferences are always transitive, excluding preferences when given explicit rules such as in Efron's dice or rock-paper-scissors ; however, aggregate preferences of a group may be intransitive. This can result in a Condorcet paradox wherein following a path from one candidate across

3657-437: The foreground, which is seen in all the details of the musical score; the middle ground, which is roughly a summary of an essential contrapuntal progression and voice-leading; and the background or Ursatz , which is one of only a few basic "long-range counterpoint" structures that are shared in the gamut of tonal music literature. Self-consciousness Self-consciousness is a heightened sense of awareness of oneself. It

3726-449: The formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false. (pp. 169–170) Hofstadter claims

3795-453: The idea behind them is the same. For example, with data structures , objects are known as nodes , superiors are called parents and subordinates are called children . In a business setting, a superior is a supervisor/boss and a peer is a colleague . Degree of branching refers to the number of direct subordinates or children an object has (in graph theory, equivalent to the number of other vertices connected to via outgoing arcs, in

3864-527: The last two levels apply to all matter , at least at the macroscopic scale . Moreover, each of these levels inherit all the properties of their children . In this particular example, there are also emergent properties —functions that are not seen at the lower level (e.g., cognition is not a property of neurons but is of the brain )—and a scalar quality (molecules are bigger than atoms, cells are bigger than molecules, etc.). Both of these concepts commonly exist in compositional hierarchies, but they are not

3933-429: The left and descending on the right. Child elements are towards the bottom of the stairs and parent elements are at the top. This structure represents hierarchical relationships through the use of visual stacking. In plain English, a hierarchy can be thought of as a set in which: The first requirement is also interpreted to mean that a hierarchy can have no circular relationships ; the association between two objects

4002-415: The more familiar Peano Arithmetic or some other sufficiently rich formal arithmetic. Thus, there are examples of sentences "which say about themselves that they are provable", but they don't exhibit the sort of downward causal powers described in the displayed quote. Hofstadter points to Bach 's Canon per Tonos , M. C. Escher 's drawings Waterfall , Drawing Hands , Ascending and Descending , and

4071-407: The problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations. A nested hierarchy or inclusion hierarchy

4140-413: The psychological "I" is a narrative fiction, something created only from intake of symbolic data and the brain's ability to create stories about itself from that data. The consequence is that a self-perspective is a culmination of a unique pattern of symbolic activity in the brain, which suggests that the pattern of symbolic activity that makes identity, that constitutes subjectivity, can be replicated within

4209-477: The rules of visual hierarchy . Visual hierarchy is also important for proper organization of files on computers. An example of visually representing hierarchy is through nested clusters. Nested clusters represent hierarchical relationships using layers of information. The child element is within the parent element, such as in a Venn diagram . This structure is most effective in representing simple hierarchical relationships. For example, when directing someone to open

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4278-501: The same direction, following a path . All parts of the hierarchy that are not linked vertically to one another nevertheless can be "horizontally" linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again. This is akin to two co-workers or colleagues ; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complementary to hierarchy. Heterarchy

4347-462: The same pattern on a screen, the latter categorizes a pattern and outputs its "essence", so that as the brain gets closer and closer to its "essence", it goes further down its strange loop. Hofstadter thinks that minds appear to determine the world by way of "downward causality ", which refers to effects being viewed in terms of their underlying causes. Hofstadter says this happens in the proof of Gödel 's incompleteness theorem : Merely from knowing

4416-469: The same property value are grouped together, and each of those resulting levels is referred to as a class . "Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity. Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to

4485-470: The size differences between the levels (but not all triangle/pyramid diagrams are hierarchical; for example, the 1992 USDA food guide pyramid ). An example of a triangle diagram appears to the right. Another common representation of a hierarchical scheme is as a tree diagram . Phylogenetic trees , charts showing the structure of § Organizations , and playoff brackets in sports are often illustrated this way. More recently, as computers have allowed

4554-562: The starting point, i.e., the original level. Examples of strange loops that Hofstadter offers include: many of the works of M. C. Escher , the Canon 5. a 2 from J.S. Bach's Musical Offering , the information flow network between DNA and enzymes through protein synthesis and DNA replication , and self-referential Gödelian statements in formal systems . In I Am a Strange Loop , Hofstadter defines strange loops as follows: And yet when I say "strange loop", I have something else in mind —

4623-399: The storage and navigation of ever larger data sets, various methods have been developed to represent hierarchies in a manner that makes more efficient use of the available space on a computer's screen. Examples include fractal maps, TreeMaps and Radial Trees . In the design field, mainly graphic design, successful layouts and formatting of the content on documents are heavily dependent on

4692-600: Was in 1881, when it was used in reference to the three orders of three angels as depicted by Pseudo-Dionysius the Areopagite (5th–6th centuries). Pseudo-Dionysius used the related Greek word (ἱεραρχία, hierarchia ) both in reference to the celestial hierarchy and the ecclesiastical hierarchy . The Greek term hierarchia means 'rule of a high priest', from hierarches (ἱεράρχης, 'president of sacred rites, high-priest') and that from hiereus (ἱερεύς, 'priest') and arche (ἀρχή, 'first place or power, rule'). Dionysius

4761-481: Was inherent in any sufficiently complex logical or arithmetical system (that allows for arithmetic by means of the Peano axioms ) in his incompleteness theorem . Gödel showed that mathematics and logic contain strange loops: propositions that not only refer to mathematical and logical truths , but also to the symbol systems expressing those truths. This leads to the sort of paradoxes seen in statements such as " This statement

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