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{{Short description|Cycles going through a hierarchy}}
{{other uses}}
{{Original research|date=June 2024}}
A '''strange loop''' is a [[Cycle (graph theory)|cyclic]] structure that goes through several levels in a hierarchical system. It arises when, by moving only upwards or downwards through the system, one finds oneself back where one started.
Strange loops may involve [[Self-reference|self-reference]] and [[Philosophy:Paradox|paradox]]. The concept of a strange loop was proposed and extensively discussed by [[Biography:Douglas Hofstadter|Douglas Hofstadter]] in ''[[Gödel, Escher, Bach]]'', and is further elaborated in Hofstadter's book ''[[Philosophy:I Am a Strange Loop|I Am a Strange Loop]]'', published in 2007.

A '''tangled hierarchy''' is a [[Hierarchy|hierarchical]] consciousness 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 is "tangled" (Hofstadter refers to this as a "[[Heterarchy|heterarchy]]"), 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, the ''Canon 5. a 2'' from J.S. Bach's Musical Offering, the information flow network between [[Biology:DNA|DNA]] and [[Philosophy:Enzymes|enzymes]] through protein synthesis and [[Biology:DNA replication|DNA replication]], and [[Self-reference|self-referential]] [[Gödel's incompleteness theorems|Gödelian statements]] in [[Formal system|formal system]]s.

In ''[[Philosophy:I Am a Strange Loop|I Am a Strange Loop]]'', Hofstadter defines strange loops as follows:

<blockquote>And yet when I say "strange loop", I have something else in mind — 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 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.&nbsp;101–102)</blockquote>

== In cognitive science ==
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 [[Biography:Kurt Gödel|Gödel]] proved was inherent in any sufficiently complex logical or arithmetical system (that allows for arithmetic by means of the [[Peano axioms]]) in his [[Gödel's incompleteness theorems|incompleteness theorem]].<ref>{{cite journal|last=Johnson|first=George|title=A New Journey into Hofstadter's Mind|journal=Scientific American|date=March 2007|volume=296|issue=3|pages=98–102|doi=10.1038/scientificamerican0307-98|bibcode=2007SciAm.296c..98J|url=http://www.scientificamerican.com/article.cfm?id=a-new-journey-into-hofsta|access-date=8 October 2011|url-access=subscription}}</ref> Gödel showed that mathematics and logic contain strange loops: propositions that not only refer to [[Logical truth|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 is false," wherein the sentence's basis of truth is found in referring to itself and its assertion, causing a logical paradox.<ref name="oreilly">{{cite journal|last=O'Reilly|first=Scott|title=I Am A Strange Loop by Douglas Hofstadter|journal=Philosophy Now|year=2010|url=http://www.philosophynow.org/issue78/I_Am_A_Strange_Loop_by_Douglas_Hofstadter|access-date=8 October 2011}}</ref>

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 [[Self-reference|twisting back upon itself]]. According to this view, 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 the brains of others, and likely even in [[Philosophy:Artificial brain|artificial brain]]s.<ref name="oreilly" />

== Strangeness ==
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 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.<ref name="Hofstadter">{{cite book|last=Hofstadter|first=Douglas|title=I Am A Strange Loop|year=2007|publisher=Basic Books |isbn=978-0-465-03078-1|url-access=registration|url=https://archive.org/details/iamstrangeloop00hofs}}</ref>

== Downward causality ==
Hofstadter thinks that minds appear to determine the world by way of "downward [[Causality|causality]]", which refers to effects being viewed in terms of their underlying causes. Hofstadter says this happens in the proof of [[Biography:Kurt Gödel|Gödel]]'s [[Gödel's incompleteness theorems|incompleteness theorem]]:
<blockquote>Merely from knowing 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)</blockquote>
Hofstadter claims a similar "flipping around of causality" appears to happen in minds possessing [[Philosophy:Self-consciousness|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,<ref>{{Cite journal |last=Nenu |first=Theodor |date=2022 |title=Douglas Hofstadter's Gödelian Philosophy of Mind |url=https://philpapers.org/rec/NENDHG |journal=Journal of Artificial Intelligence and Consciousness|volume=9 |issue=2 |pages=241–266 |doi=10.1142/S2705078522500011 }}</ref> together with other aspects of Hofstadter's metaphysics of mind. Nenu also questions the correctness of 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 [[Biography:Leon Henkin|Leon Henkin]]). It turns out that under suitable [[Metamathematics|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 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.

== Examples ==
Hofstadter points to Bach's ''Canon per Tonos'', M. C. Escher's drawings ''[[Unsolved:Waterfall (M. C. Escher)|Waterfall]]'', ''Drawing Hands'', ''[[Unsolved:Ascending and Descending|Ascending and Descending]]'', and the [[Philosophy:Liar paradox|liar paradox]] as examples that illustrate the idea of strange loops, which is expressed fully in the proof of Gödel's [[Gödel's incompleteness theorems|incompleteness theorem]].

The "[[Chicken or the egg|chicken or the egg]]" paradox is perhaps the best-known strange loop problem.

The "[[Ouroboros|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 is another illustrative example of a strange loop. Named after [[Biography:Roger Shepard|Roger Shepard]], it is a [[Physics:Sound|sound]] consisting of a superposition of tones separated by [[Octave|octave]]s. When played with the base [[Physics:Pitch (music)|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 a similar way a sound with seemingly ever increasing tempo can be constructed, as was demonstrated by Jean-Claude Risset.
{{listen|filename=DescenteInfinie.ogg|title=A Shepard–Risset glissando|description=|format=[[Software:Ogg|Ogg]]}}

Visual illusions depicting strange loops include the [[Physics:Penrose stairs|Penrose stairs]] and the [[Physics:Barberpole illusion|Barberpole illusion]].

A [[Quine (computing)|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|metamorphic code]].

[[Intransitive dice#Efron's dice|Efron's dice]] are four dice that are [[Intransitivity|intransitive]] under gambler's preference. I.e., the dice are ordered {{nowrap|A > B > C > D > A}}, where {{nowrap|''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 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 [[Philosophy:Polysemy|polysemy]] 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'' is an old Japanese [[Social:Fairy tale|fairy tale]] with a story that explains social and natural hierarchies as a strange loop.

A strange loop can be found by traversing the links in the “See also” sections of the respective English Wikipedia articles. For instance: This article->Mise en abyme->[[Recursion]]->this article.<ref>Wikipedia contributors. (2024, December 14). Strange loop. In ''Wikipedia, The Free Encyclopedia''. Retrieved 10:34, December 25, 2024, from https://en.wikipedia.org/w/index.php?title=Strange_loop&oldid=1263113776</ref>{{Circular reference|date=December 2024}}

== See also ==
* ''A Strange Loop'' – A Broadway musical by Michael R. Jackson that takes its title from and references Hofstadter's strange loop
* {{annotated link|Philosophy:Absurdism}}
* {{annotated link|Autopoiesis}}
* {{annotated link|Physics:Causal loop}}
* {{annotated link|Philosophy:Dilemma}}
* {{annotated link|Philosophy:Euthyphro dilemma}}
* {{annotated link|Physics:Grandfather paradox}} – Going back in time to kill one's own grandfather generates a circular contradiction
* {{annotated link|Social:Hysteron proteron}}
* {{annotated link|Klein bottle}}
* {{annotated link|Philosophy:Meno}} – Paradox: One must already possess any given piece of knowledge, otherwise it could not be recognized when supposedly "discovered"
* {{annotated link|Metamorphic code}}
* {{annotated link|Möbius strip}}
* {{annotated link|Philosophy:Münchhausen trilemma}}
* {{annotated link|Ouroboros}}
* {{annotated link|Physics:Penrose stairs}}
* {{annotated link|Unsolved:Perpetual motion}}
* {{annotated link|Unsolved:Phoenix (mythology)}}
* {{annotated link|Polytely}}
* {{annotated link|Social:Reflexivity (social theory)}}
* {{annotated link|Rock paper scissors}}
* {{annotated link|Three hares}}
* {{annotated link|Tupper's self-referential formula}}

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
*{{cite journal |last=Tanenbaum |first=P. J. |date=October 1999 |title= Simultaneous intersection representation of pairs of graphs |journal = Journal of Graph Theory |volume=32 |issue=2 |pages=171–190 |doi= 10.1002/(SICI)1097-0118(199910)32:2<171::AID-JGT7>3.0.CO;2-N |issn = 1097-0118}} *{{cite journal |last=Nenu |first=T. |date=September 2022 |title= Douglas Hofstadter's Gödelian Philosophy of Mind |journal = Journal of Artificial Intelligence and Consciousness |volume=9 |issue=2 |pages=241–266 |doi= 10.1142/S2705078522500011 |issn = 2705-0793|doi-access=free |hdl=1983/07af725f-5af7-44f5-9d9f-df197218c741 |hdl-access=free }} {{refend}}

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{{Douglas Hofstadter}}

[[Category:Hierarchy]]
[[Category:Paradoxes]]
[[Category:Philosophical analogies]]

{{Sourceattribution|Strange loop}}