Strange loop

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A strange loop is a 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 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 , published in 2007.

Contents

A tangled hierarchy is a 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"), 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 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 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. 101–102)

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 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. [1] Gödel showed that mathematics and logic contain strange loops: propositions that not only refer to 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. [2]

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 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 artificial brains. [2]

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. [3]

Downward causality

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. 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" appears to happen in minds possessing 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, [4] 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 Leon Henkin). It turns out that under suitable 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 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 "chicken or the egg" paradox is perhaps the best-known strange loop problem.

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.

A Shepard tone is another illustrative example of a strange loop. Named after Roger Shepard, it is a sound consisting of a superposition of tones 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. In a similar way a sound with seemingly ever increasing tempo can be constructed, as was demonstrated by Jean-Claude Risset.

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A Shepard–Risset glissando
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Visual illusions depicting strange loops include the Penrose stairs and the Barberpole illusion.

A 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.

Efron's dice are four dice that are intransitive under gambler's preference. I.e., the dice are ordered A > B > C > D > A, where 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 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 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. [5] [ circular reference ]

See also

Related Research Articles

<span class="mw-page-title-main">Douglas Hofstadter</span> American professor of cognitive science (born 1945)

Douglas Richard Hofstadter is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, strange loops, artificial intelligence, and discovery in mathematics and physics. His 1979 book Gödel, Escher, Bach: An Eternal Golden Braid won the Pulitzer Prize for general nonfiction, and a National Book Award for Science. His 2007 book I Am a Strange Loop won the Los Angeles Times Book Prize for Science and Technology.

<span class="mw-page-title-main">Gödel's completeness theorem</span> Fundamental theorem in mathematical logic

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

<span class="mw-page-title-main">Self-reference</span> Sentence, idea or formula that refers to itself

Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.

<i>Gödel, Escher, Bach</i> 1979 book by Douglas Hofstadter

Gödel, Escher, Bach: an Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter.

<span class="mw-page-title-main">Shepard tone</span> Auditory illusion

A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the bass 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 seems to continually ascend or descend in pitch, yet which ultimately gets no higher or lower.

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

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<span class="mw-page-title-main">Metamathematics</span> Study of mathematics itself

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<span class="mw-page-title-main">Egbert B. Gebstadter</span> Fictional author

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The philosophy of artificial intelligence is a branch of the philosophy of mind and the philosophy of computer science that explores artificial intelligence and its implications for knowledge and understanding of intelligence, ethics, consciousness, epistemology, and free will. Furthermore, the technology is concerned with the creation of artificial animals or artificial people so the discipline is of considerable interest to philosophers. These factors contributed to the emergence of the philosophy of artificial intelligence.

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<i>I Am a Strange Loop</i> 2007 book by Douglas Hofstadter

I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop to explain the sense of "I". The concept of a strange loop was originally developed in his 1979 book Gödel, Escher, Bach.

In the end, we are self-perceiving, self-inventing, locked-in mirages that are little miracles of self-reference.

<i>Print Gallery</i> (M. C. Escher) Lithograph printed in 1956 by the Dutch artist M. C. Escher

Print Gallery is a lithograph printed in 1956 by the Dutch artist M. C. Escher. It depicts a man in a gallery viewing a print of a seaport, and among the buildings in the seaport is the very gallery in which he is standing, making use of the Droste effect with visual recursion. The lithograph has attracted discussion in both mathematical and artistic contexts. Escher considered Print Gallery to be among the best of his works.

Mechanism is the belief that natural wholes are similar to complicated machines or artifacts, composed of parts lacking any intrinsic relationship to each other.

The Penrose–Lucas argument is a logical argument partially based on a theory developed by mathematician and logician Kurt Gödel. In 1931, he proved that every effectively generated theory capable of proving basic arithmetic either fails to be consistent or fails to be complete. Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing Machine that works on Peano arithmetic because the latter can't see the truth value of its Gödel sentence, while human minds can. Mathematician Roger Penrose modified the argument in his first book on consciousness, The Emperor's New Mind (1989), where he used it to provide the basis of his theory of consciousness: orchestrated objective reduction.

References

Citations

  1. ↑ Johnson, George (March 2007). "A New Journey into Hofstadter's Mind". Scientific American. 296 (3): 98–102. Bibcode:2007SciAm.296c..98J. doi:10.1038/scientificamerican0307-98 . Retrieved 8 October 2011.
  2. 1 2 O'Reilly, Scott (2010). "I Am A Strange Loop by Douglas Hofstadter". Philosophy Now. Retrieved 8 October 2011.
  3. ↑ Hofstadter, Douglas (2007). I Am A Strange Loop . Basic Books. ISBN   978-0-465-03078-1.
  4. ↑ Nenu, Theodor (2022). "Douglas Hofstadter's Gödelian Philosophy of Mind". Journal of Artificial Intelligence and Consciousness. 9 (2): 241–266. doi:10.1142/S2705078522500011.
  5. ↑ 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

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