Strange loop

M. C. Escher − Drawing Hands, 1948.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.

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. ”

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