Strange loop

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.

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 was inherent in any sufficiently complex logical or arithmetical system (that allows for arithmetic by means of the Peano axioms) in his incompleteness theorem.

[3] 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 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.

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 Leon Henkin).

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 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 "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.

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.

The Stonecutter is an old Japanese 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.