Strange loop

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A strange loop arises when, by moving up or down through a hierarchical system, one finds oneself back where one started.

Strange loops may involve self-reference and paradox. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter in Gödel, Escher, Bach, and is further elaborated in Hofstadter's book I Am a Strange Loop, which appeared in 2007.

A tangled hierarchy is a hierarchical system in which a strange loop appears.


[edit] Definitions

A strange loop is a hierarchy of levels, each of which may consist of objects, processes, or virtually anything else (such is the generality of the notion). Each level is linked to at least one other by some type of relationship. A strange loop hierarchy, however, is "tangled" (what Hofstadter refers to as a "heterarchy"), in that there is no well defined highest or lowest level. The levels are organized such that moving through them eventually returns one to one's starting point, i.e., the original level. Examples of strange loops that Hofstadter offers include many of the works of M. C. Escher, 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 — 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 a 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. 101-102)

[edit] Strangeness

The "strangeness" of a strange loop comes from Hofstadter's idea of "downward causality", which refers to situation where a cause-and-effect relationship in a system gets flipped upside-down. Hofstadter claims this happens in the proof of Gödel's Incompleteness Theorem:

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)

Hofstadter claims a similar "flipping around of causality" happens in minds possessing self-consciousness. 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.

[edit] Examples

  • A Shepard tone is another illustrative example of a strange loop. Named after Roger Shepard, it is a sound consisting of a superposition of sine waves 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
  • Quines in software programing- wherein a program produces a new version of itself without any input from the outside. Metamorphic code.
  • The mathematical phenomenon of 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).

[edit] Strange loops in popular culture

  • The myriad time-travel paradoxes of classic Science Fiction can be perceived as creative versions of Strange Loops that fail by self-cancelling feedback. The two base forms of this are the Predestination paradox and the Ontological paradox
Usually, these stories are variations on a core trope: a time-traveler alters events in the past in a way that precludes the initiation of the whole process in the first place - either by rendering time-travel itself impossible, or by somehow rendering his own existence null and void. Archetypal examples would include almost any of the many “Shoot Your Grandfather” stories (cf, "The Men Who Murdered Mohammed" by Alfred Bester).
A quintessential (and recognized classic) SF Time-Travel Strange Loop story is “—All You Zombies—” by Robert A. Heinlein. In this short story, Heinlein creates an Anti-Paradox (a self-generating/self-supporting/self-dependent tautological sequence) that exactly meets Douglas Hofstadter's criteria for a Strange Loop as quoted in Definitions (above) - specifically in regard to the criteria ...
"... 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.
The recursive loop in “—All You Zombies—” occurs by way of an alteration of the past that leads to the very future in which same alteration becomes both necessary and inevitable. Specifically: a young hermaphrodite in female mode (as surgically adjusted right after birth so as to render her female reproductive organs dominant and fertile) becomes her own mother (supplying the egg and the womb) after being impregnated by her older time-travelling male version (after giving birth she/he is surgically adjusted into fertile and active male mode). This time-looped self-encounter/courtship/impregnation is stage-managed by her/his/their much older, male self who brings the younger male version back in time to meet (and court, seduce and impregnate) the female iteration. This all to ensure his/her/their eternal, self-generating closed-loop existence.
nota bene: during the story, the song "I'm My Own Grandpa" cited in Examples (above) is played on the jukebox in the bar managed by the narrator - who is the "final edition" of the main character.

[edit] References

  • P.J. Tanenbaum, "Simultaneous intersection representation of pairs of graphs," Journal of Graph Theory 32 (1999) 171-190. ISSN 1097-0118

[edit] See also

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