Strange loop
Encyclopedia
Strange loops may involve self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
and paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter
Douglas Hofstadter
Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...
in Gödel, Escher, Bach
Gödel, Escher, Bach
Gödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....
, and is further elaborated in Hofstadter's book I Am a Strange Loop
I Am a Strange Loop
I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop originally developed in his 1979 book Gödel, Escher, Bach....
, published in 2007.
A tangled hierarchy is a hierarchical
Hierarchy
A hierarchy is an arrangement of items in which the items are represented as being "above," "below," or "at the same level as" one another...
system in which a strange loop appears.
Definitions
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, however, is "tangled" (Hofstadter refers to this as a "heterarchyHeterarchy
A heterarchy is a system of organization replete with overlap, multiplicity, mixed ascendancy, and/or divergent-but-coexistent patterns of relation...
"), in that there is no well defined highest or lowest level; moving through the levels one eventually returns to the starting point, i.e., the original level. Examples of strange loops that Hofstadter offers include: many of the works of M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
, the information flow network between DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...
and enzymes through protein synthesis and DNA replication
DNA replication
DNA replication is a biological process that occurs in all living organisms and copies their DNA; it is the basis for biological inheritance. The process starts with one double-stranded DNA molecule and produces two identical copies of the molecule...
, and self-referential
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
Gödelian statements
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
in formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
s.
In I Am a Strange Loop
I Am a Strange Loop
I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop originally developed in his 1979 book Gödel, Escher, Bach....
, Hofstadter defines strange loops as follows:
Strange loops in cognitive science
Strange loops take form in human consciousness as the complexity of active symbols in the brain inevitably lead to the same kind of self-reference which Gödel proved was inherent in any complex logical or arithmetical system in his Incompleteness Theorem. Gödel showed that mathematics and logic contain Strange Loops: propositions that not only refer to mathematical/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 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. We are not born with an ‘I’ – the ego emerges only gradually as experience shapes our dense web of active symbols into a tapestry rich and complex enough to begin twisting back upon itself
Self-fulfilling prophecy
A self-fulfilling prophecy is a prediction that directly or indirectly causes itself to become true, by the very terms of the prophecy itself, due to positive feedback between belief and behavior. Although examples of such prophecies can be found in literature as far back as ancient Greece and...
. According to this view the psychological ‘I’ is a narrative fiction, something created only from intake of symbolic data and its own ability to create stories about itself from that data. The consequence is that a perspective (a mind) is a culmination of a unique pattern of symbolic activity in our nervous systems, which suggests that the pattern of symbolic activity that makes identity, that constitutes subjectivity, can be replicated within the brains of others, and perhaps even in artificial brains.
Strangeness
The "strangeness" of a strange loop comes from our way of perception; because we categorize our input in a small amount of 'symbols' (by which Hofstadter means groups of neurons standing for one thing in the outside world). So the difference between the video-feedback loop and our strange loops, our "I"'s, is that while the former one converts light to the same pattern on a screen, the latter one categorizes a pattern and outputs its essence, so that as you get closer and closer to your essence the further you get down your strange loop (according to Hofstadter's book 'I am a Strange Loop').Downward causality
Hofstadter thinks our minds appear to us to determine the world by way of "downward causalityCausality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
", which refers to a situation where a cause-and-effect relationship in a system gets flipped upside-down. Hofstadter claims this happens in the proof of Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
's Incompleteness Theorem
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
:
Hofstadter claims a similar "flipping around of causality" appears to happen in minds possessing self-consciousness
Self-consciousness
Self-consciousness is an acute sense of self-awareness. It is a preoccupation with oneself, as opposed to the philosophical state of self-awareness, which is the awareness that one exists as an individual being; although some writers use both terms interchangeably or synonymously...
. The mind perceives itself as the cause of certain feelings, ("I" am the source of my desires), while scientifically, feelings and desires are strictly caused by the interactions of neurons, and ultimately, the probabilistic laws of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
Examples
Hofstadter points to BachJohann Sebastian Bach
Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
's Canon per Tonos, M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
's drawings Waterfall
Waterfall (M. C. Escher)
Waterfall is a lithography print by the Dutch artist M. C. Escher which was first printed in October, 1961. It shows an apparent paradox where water from the base of a waterfall appears to run downhill before reaching the top of the waterfall....
, Drawing Hands
Drawing Hands
Drawing Hands is a lithograph by the Dutch artist M. C. Escher first printed in January 1948. It depicts a sheet of paper out of which, from wrists that remain flat on the page, two hands rise, facing each other and in the paradoxical act of drawing one another into existence...
, Ascending and Descending
Ascending and Descending
Ascending and Descending is a lithograph print by the Dutch artist M. C. Escher which was first printed in March 1960.The original print measures 14" x 11 1/4”. The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identically dressed men appear on the staircase,...
, and the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
as examples that illustrate the idea of strange loops, which is expressed fully in the proof of Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
's incompleteness theorem
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
.
A Shepard tone
Shepard tone
A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the base pitch of the tone moving upward or downward, it is referred to as the Shepard scale. This creates the auditory illusion of a tone that continually...
is another illustrative example of a strange loop. Named after Roger Shepard
Roger Shepard
Roger Newland Shepard is a cognitive scientist and author of Toward a Universal Law of Generalization for Psychological Science. He is seen as a father of research on spatial relations....
, it is a sound
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
consisting of a superposition of sine waves separated by octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
s. When played with the base pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
of the tone moving upwards or downwards, it is referred to as the Shepard scale. This creates the auditory illusion
Auditory illusion
An auditory illusion is an illusion of hearing, the aural equivalent of an optical illusion: the listener hears either sounds which are not present in the stimulus, or "impossible" sounds...
of a tone that continually ascends or descends in pitch, yet which ultimately seems to get no higher or lower. See Barber's pole
Barber's pole
A barber's pole is a type of sign used by barbers to signify the place or shop where they perform their craft. The trade sign is, by a tradition dating back to the Middle Ages, a staff or pole with a helix of colored stripes...
.
Quines in software programming, wherein a program produces a new version of itself without any input from the outside. Metamorphic code
Metamorphic code
In computer virus terms, metamorphic code is code that can reprogram itself. Often, it does this by translating its own code into a temporary representation, editing the temporary representation of itself, and then writing itself back to normal code again. This procedure is done with the virus...
.
Efron's dice are four dice, each of which has a high probability of beating one of the others.
The liar paradox and Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
also involve strange loops as does René Magritte's
René Magritte
René François Ghislain Magritte[p] was a Belgian surrealist artist. He became well known for a number of witty and thought-provoking images...
painting The Treachery of Images
The Treachery Of Images
The Treachery of Images is a painting by the Belgian René Magritte, painted when Magritte was 30 years old. The picture shows a pipe...
.
The mathematical phenomenon of polysemy
Polysemy
Polysemy is the capacity for a sign or signs to have multiple meanings , i.e., a large semantic field.Charles Fillmore and Beryl Atkins’ definition stipulates three elements: the various senses of a polysemous word have a central origin, the links between these senses form a network, and ...
has been observed to be 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
The Stonecutter
The Stonecutter is a Chinese folklore of unknown authorship. It is closely related to the themes of The Fisherman and His Wife, a well known fairy tale collected by the Brothers Grimm. In the legend, a poor stonecutter takes notice of the lifestyles of those from higher in the social hierarchy and...
is an old Japanese fairy tale
Fairy tale
A fairy tale is a type of short story that typically features such folkloric characters, such as fairies, goblins, elves, trolls, dwarves, giants or gnomes, and usually magic or enchantments. However, only a small number of the stories refer to fairies...
with a story that explains social and natural hierarchies as a strange loop.
The "Chicken and the Egg" paradox is perhaps the most well-known strange loop problem.
See also
- AbsurdAbsurdAbsurd or The Absurd may refer to:* Absurdity, general and technical usage - associated with extremely poor reasoning, the ridiculous, or nonsense....
- AbsurdismAbsurdismIn philosophy, "The Absurd" refers to the conflict between the human tendency to seek value and meaning in life and the human inability to find any...
- AbsurdityAbsurdityAn absurdity is a thing that is extremely unreasonable, so as to be foolish or not taken seriously, or the state of being so. "Absurd" is an adjective used to describe an absurdity, e.g., “this encyclopedia article is absurd”. It derives from the Latin absurdusm meaning "out of tune", hence...
- Autopoiesis
- ConundrumConundrumConundrum may refer to:* A riddle whose answer is or involves a pun or unexpected twist* A logical postulation that evades resolution, an intricate and difficult problem- Literature :...
- DilemmaDilemmaA dilemma |proposition]]") is a problem offering two possibilities, neither of which is practically acceptable. One in this position has been traditionally described as "being on the horns of a dilemma", neither horn being comfortable...
- EnigmaEnigmaAn enigma is a type of riddle generally expressed in radical or allegorical language that requires ingenuity and careful thought for its solution.Enigma, aenigma, or enigmatic may also refer to:-Music:...
- Euthyphro dilemmaEuthyphro dilemmaThe Euthyphro dilemma is found in Plato's dialogue Euthyphro, in which Socrates asks Euthyphro: "Is the pious loved by the gods because it is pious, or is it pious because it is loved by the gods?"...
- Feedback loopFeedbackFeedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
- IronyIronyIrony is a rhetorical device, literary technique, or situation in which there is a sharp incongruity or discordance that goes beyond the simple and evident intention of words or actions...
- Klein bottleKlein bottleIn mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
- Möbius stripMöbius stripThe Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
- Ontological paradox
- Optical feedbackOptical feedbackOptical feedback is the optical equivalent of acoustic feedback. A simple example is the feedback that occurs when a loop exists between an optical input, e.g., a video camera, and an optical output, e.g., a television screen or monitor...
- ParadoxParadoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
- Pitch circularityPitch circularityPitch is often defined as extending along a one-dimensional continuum from high to low, as can be experienced by sweeping one’s hand up or down a piano keyboard. This continuum is known as pitch height...
- PolytelyPolytelyPolytely can be described as frequently, complex problem-solving situations characterized by the presence of not one, but several goals, endings.Modern societies face an increasing incidence of various complex problems...
- Predestination paradoxPredestination paradoxA predestination paradox is a paradox of time travel that is often used as a convention in science fiction. It exists when a time traveller is caught in a loop of events that "predestines" or "predates" them to travel back in time...
- Quine
- Shepard toneShepard toneA Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the base pitch of the tone moving upward or downward, it is referred to as the Shepard scale. This creates the auditory illusion of a tone that continually...
- Three haresThree haresThe three hares is a circular motif appearing in sacred sites from the Middle and Far East to the churches of southwest England , and historical synagogues in Europe....