Douglas Hofstadter

Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world,[1][4] consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. His 1979 book Gödel, Escher, Bach: An Eternal Golden Braid won both the Pulitzer Prize for general nonfiction[5][6] and a National Book Award (at that time called The American Book Award) for Science.[7][note 1] His 2007 book I Am a Strange Loop won the Los Angeles Times Book Prize for Science and Technology.[8][9][10]

Douglas Hofstadter
Hofstadter in Bologna, Italy, in 2002
Born
Douglas Richard Hofstadter

(1945-02-15) February 15, 1945
New York City, United States
NationalityAmerican
EducationStanford University (BSc)
University of Oregon (PhD, 1974)
Known forGödel, Escher, Bach
I Am a Strange Loop[1]
Hofstadter's butterfly
Hofstadter's law
Spouse(s)Carol Ann Brush (1985–1993; her death; 2 children)
Baofen Lin (2012–present)
AwardsNational Book Award
Pulitzer Prize
Member of the American Academy of Arts and Sciences
Golden Plate Award of the American Academy of Achievement[2]
Scientific career
FieldsCognitive science
Philosophy of mind
Translation
Physics
InstitutionsIndiana University
Stanford University
University of Oregon
University of Michigan
ThesisThe Energy Levels of Bloch Electrons in a Magnetic Field (1974)
Doctoral advisorGregory Wannier[3]
Doctoral studentsDavid Chalmers
Robert M. French
Scott A. Jones
Melanie Mitchell
Websiteprelectur.stanford.edu/lecturers/hofstadter

Early life and education

Hofstadter was born in New York City to Jewish parents: Nobel Prize-winning physicist Robert Hofstadter and Nancy Givan Hofstadter.[11] He grew up on the campus of Stanford University, where his father was a professor, and attended the International School of Geneva in 1958–59. He graduated with Distinction in mathematics from Stanford University in 1965, and received his Ph.D. in physics[3][12] from the University of Oregon in 1975, where his study of the energy levels of Bloch electrons in a magnetic field led to his discovery of the fractal known as Hofstadter's butterfly.[12]

Academic career

Since 1988, Hofstadter has been the College of Arts and Sciences Distinguished Professor of Cognitive Science and Comparative Literature at Indiana University in Bloomington, where he directs the Center for Research on Concepts and Cognition which consists of himself and his graduate students, forming the "Fluid Analogies Research Group" (FARG).[13] He was initially appointed to the Indiana University's Computer Science Department faculty in 1977, and at that time he launched his research program in computer modeling of mental processes (which he called "artificial intelligence research", a label he has since dropped in favor of "cognitive science research"). In 1984, he moved to the University of Michigan in Ann Arbor, where he was hired as a professor of psychology and was also appointed to the Walgreen Chair for the Study of Human Understanding. In 1988 he returned to Bloomington as "College of Arts and Sciences Professor" in both cognitive science and computer science. He was also appointed adjunct professor of history and philosophy of science, philosophy, comparative literature, and psychology, but has said that his involvement with most of those departments is nominal.[14][15][16] In 1988 Hofstadter received the In Praise of Reason award, the Committee for Skeptical Inquiry's highest honor.[17] In April 2009 he was elected a Fellow of the American Academy of Arts and Sciences[18] and a member of the American Philosophical Society.[19] In 2010 he was elected a member of the Royal Society of Sciences in Uppsala, Sweden.[20]

Hofstadter's many interests include music, visual art, the mind, creativity, consciousness, self-reference, translation and mathematics.

Hofstadter giving a presentation at the 2006 Singularity Summit

At the University of Michigan and Indiana University, he and Melanie Mitchell coauthored a computational model of "high-level perception"—Copycat—and several other models of analogy-making and cognition, including the Tabletop project, co-developed with Robert M. French. Hofstadter's doctoral student James Marshall subsequently extended the Copycat project under the name "Metacat".[21] The Letter Spirit project, implemented by Gary McGraw and John Rehling, aims to model artistic creativity by designing stylistically uniform "gridfonts" (typefaces limited to a grid). Other more recent models include Phaeaco (implemented by Harry Foundalis) and SeqSee (Abhijit Mahabal), which model high-level perception and analogy-making in the microdomains of Bongard problems and number sequences, respectively, as well as George (Francisco Lara-Dammer), which models the processes of perception and discovery in triangle geometry.[22][23][24]

The pursuit of beauty has driven Hofstadter both inside and outside his professional work. He seeks beautiful mathematical patterns, beautiful explanations, beautiful typefaces, beautiful sonic patterns in poetry, etc. Hofstadter has said of himself, "I'm someone who has one foot in the world of humanities and arts, and the other foot in the world of science." He has had several exhibitions of his artwork in various university galleries. These shows have featured large collections of his gridfonts, his ambigrams (pieces of calligraphy created with two readings, either of which is usually obtained from the other by rotating or reflecting the ambigram, but sometimes simply by "oscillation", like the Necker Cube or the rabbit/duck figure of Joseph Jastrow), and his "Whirly Art" (music-inspired visual patterns realized using shapes based on various alphabets from India). Hofstadter invented the term "ambigram" in 1984; many ambigrammists have since taken up the concept.[25]

Hofstadter collects and studies cognitive errors (largely, but not solely, speech errors), "bon mots" (spontaneous humorous quips), and analogies of all sorts, and his longtime observation of these diverse products of cognition, and his theories about the mechanisms that underlie them, have exerted a powerful influence on the architectures of the computational models he and FARG members have developed.[26]

All FARG computational models share certain key principles, including:

  • that human thinking is carried out by thousands of independent small actions in parallel, biased by the concepts that are currently activated
  • that activation spreads from activated concepts to less activated "neighbor concepts"
  • that there is a "mental temperature" that regulates the degree of randomness in the parallel activity
  • that promising avenues tend to be explored more rapidly than unpromising ones

FARG models also have an overarching philosophy that all cognition is built from the making of analogies. The computational architectures that share these precepts are called "active symbols" architectures.

Hofstadter's thesis about consciousness, first expressed in Gödel, Escher, Bach (GEB) but also present in several of his later books, is that it is an emergent consequence of seething lower-level activity in the brain. In GEB he draws an analogy between the social organization of a colony of ants and the mind seen as a coherent "colony" of neurons. In particular, Hofstadter claims that our sense of having (or being) an "I" comes from the abstract pattern he terms a "strange loop", an abstract cousin of such concrete phenomena as audio and video feedback that Hofstadter has defined as "a level-crossing feedback loop". The prototypical example of a strange loop is the self-referential structure at the core of Gödel's incompleteness theorems. Hofstadter's 2007 book I Am a Strange Loop carries his vision of consciousness considerably further, including the idea that each human "I" is distributed over numerous brains, rather than being limited to one.[27]

Hofstadter's writing is characterized by an intense interaction between form and content, as exemplified by the 20 dialogues in GEB, many of which simultaneously discuss and imitate strict musical forms used by Bach, such as canons and fugues. Most of Hofstadter's books feature some kind of structural alternation: in GEB between dialogues and chapters, in The Mind's I between selections and reflections, in Metamagical Themas between Chapters and Postscripts, and so forth. In both his writing and his teaching, Hofstadter stresses the concrete, constantly using examples and analogies, and avoids the abstract. Typical of the courses he teaches is his seminar "Group Theory and Galois Theory Visualized", in which abstract mathematical ideas are rendered as concretely as possible. He puts great effort into making ideas clear and visual, and asserts that when he teaches, if his students do not understand something, it is never their fault but always his own.

Hofstadter is passionate about languages. In addition to English, his mother tongue, he speaks French and Italian fluently (the language spoken at home with his children is Italian). At various times in his life, he has studied (in descending order of level of fluency reached) German, Russian, Spanish, Swedish, Mandarin, Dutch, Polish, and Hindi. His love of sounds pushes him to strive to minimize, and ideally get rid of, any foreign accent.

Le Ton beau de Marot: In Praise of the Music of Language is a long book devoted to language and translation, especially poetry translation, and one of its leitmotifs is a set of 88 translations of "Ma Mignonne", a highly constrained poem by 16th-century French poet Clément Marot. In this book, Hofstadter jokingly describes himself as "pilingual" (meaning that the sum total of the varying degrees of mastery of all the languages that he has studied comes to 3.14159 ...), as well as an "oligoglot" (someone who speaks "a few" languages).[28][29]

In 1999, the bicentennial year of the Russian poet and writer Alexander Pushkin, Hofstadter published a verse translation of Pushkin's classic novel-in-verse Eugene Onegin. He has translated many other poems, too (always respecting their formal constraints) and two novels (in prose): La Chamade (That Mad Ache) by French writer Françoise Sagan, and La Scoperta dell'Alba (The Discovery of Dawn) by Walter Veltroni, the then-head of the Partito Democratico in Italy. The Discovery of Dawn was published in 2007, and That Mad Ache was published in 2009, bound together with Hofstadter's essay Translator, Trader: An Essay on the Pleasantly Pervasive Paradoxes of Translation.

Hofstadter's Law
Main article: Hofstadter's Law

Hofstadter's Law is "It always takes longer than you expect, even when you take into account Hofstadter's Law." The law is stated in GEB.

Students

Hofstadter's former Ph.D. students[30] include (with dissertation title):

  • Donald Byrd—Music Notation by Computer
  • David Chalmers—Toward a Theory of Consciousness
  • Gray A. Clossman—A Model of Categorization and Learning in a Connectionist Broadcast System
  • Hamid Ekbia—AI Dreams and Discourse: Science and Engineering in Tension
  • Harry Foundalis—Phaeaco: A Cognitive Architecture Inspired by Bongard Problems
  • Bob French—Tabletop: An Emergent, Stochastic Model of Analogy-Making
  • Francisco Lara-Dammer—Modeling Human Discoverativity in Geometry
  • Abhijit Mahabal—SeqSee: A Concept-centered Architecture for Sequence Perception
  • Jim Marshall—Metacat: A Self-Watching Cognitive Architecture for Analogy-making and High level Perception
  • Gary McGraw—Letter Spirit (Part One): Emergent High-level Perception of Letters Using Fluid Concepts[31]
  • Marsha Meredith—Seek-Whence: A Model of Pattern Perception
  • Eric Nichols—Musicat: A Computer Model of Musical Listening and Analogy-Making
  • Melanie Mitchell—Copycat: A Computer Model of High-Level Perception and Conceptual Slippage in Analogy-making
  • John Rehling—Letter Spirit (Part Two): Modeling Creativity in a Visual Domain
  • Wang Pei (Pei Wang)—Non-Axiomatic Reasoning System: Exploring the Essence of Intelligence
  • William York—Aesthetics and the Scope and Limits of Cognitive Science

Public image

Hofstadter has said that he feels "uncomfortable with the nerd culture that centers on computers". He admits that "a large fraction [of his audience] seems to be those who are fascinated by technology", but when it was suggested that his work "has inspired many students to begin careers in computing and artificial intelligence" he replied that he was pleased about that, but that he himself has "no interest in computers".[32][33] In that interview he also mentioned a course he has twice given at Indiana University, in which he took a "skeptical look at a number of highly touted AI projects and overall approaches".[16] For example, upon the defeat of Garry Kasparov by Deep Blue, he commented that "It was a watershed event, but it doesn't have to do with computers becoming intelligent".[34] Hofstadter's disinterest in computers is analogous to an astronomer's disinterest in telescopes. In his book Metamagical Themas, he says that "in this day and age, how can anyone fascinated by creativity and beauty fail to see in computers the ultimate tool for exploring their essence?".[35]

Provoked by predictions of a technological singularity (a hypothetical moment in the future of humanity when a self-reinforcing, runaway development of artificial intelligence causes a radical change in technology and culture), Hofstadter has both organized and participated in several public discussions of the topic. At Indiana University in 1999 he organized such a symposium, and in April 2000, he organized a larger symposium titled "Spiritual Robots" at Stanford University, in which he moderated a panel consisting of Ray Kurzweil, Hans Moravec, Kevin Kelly, Ralph Merkle, Bill Joy, Frank Drake, John Holland and John Koza. Hofstadter was also an invited panelist at the first Singularity Summit, held at Stanford in May 2006. Hofstadter expressed doubt that the singularity will occur in the foreseeable future.[36][37][38][39][40][41]

In 1988 Dutch director Piet Hoenderdos created a docudrama about Hofstadter and his ideas, Victim of the Brain, based on The Mind's I. It includes interviews with Hofstadter about his work.[42]

Columnist

When Martin Gardner retired from writing his "Mathematical Games" column for Scientific American magazine, Hofstadter succeeded him in 1981–83 with a column titled Metamagical Themas (an anagram of "Mathematical Games"). An idea he introduced in one of these columns was the concept of "Reviews of This Book", a book containing nothing but cross-referenced reviews of itself that has an online implementation.[43] One of Hofstadter's columns in Scientific American concerned the damaging effects of sexist language, and two chapters of his book Metamagical Themas are devoted to that topic, one of which is a biting analogy-based satire, "A Person Paper on Purity in Language" (1985), in which the reader's presumed revulsion at racism and racist language is used as a lever to motivate an analogous revulsion at sexism and sexist language; Hofstadter published it under the pseudonym William Satire, an allusion to William Safire.[44] Another column reported on the discoveries made by University of Michigan professor Robert Axelrod in his computer tournament pitting many iterated prisoner's dilemma strategies against each other, and a follow-up column discussed a similar tournament that Hofstadter and his graduate student Marek Lugowski organized. The "Metamagical Themas" columns ranged over many themes, including patterns in Frédéric Chopin's piano music (particularly his études), the concept of superrationality (choosing to cooperate when the other party/adversary is assumed to be equally intelligent as oneself), and the self-modifying game of Nomic, based on the way in which the legal system modifies itself, and developed by philosopher Peter Suber.[45]

Personal life

Hofstadter was married to Carol Ann Brush until her death. They met in Bloomington, and married in Ann Arbor in 1985. They had two children, Danny and Monica. Carol died in 1993 from the sudden onset of a brain tumor — glioblastoma multiforme — when their children were 5 and 2. The Carol Ann Brush Hofstadter Memorial Scholarship for Bologna-bound Indiana University students was established in 1996 in her name.[46] Hofstadter's book Le Ton beau de Marot is dedicated to their two children and its dedication reads "To M. & D., living sparks of their Mommy's soul".

In the fall of 2010, Hofstadter met Baofen Lin in a cha-cha-cha class, and they married in Bloomington in September 2012.[47]

Hofstadter has composed numerous pieces for piano and a few for piano and voice. He created an audio CD, DRH/JJ, which includes all these compositions performed mostly by pianist Jane Jackson, with a few performed by Brian Jones, Dafna Barenboim, Gitanjali Mathur and Hofstadter.[48]

The dedication for I Am A Strange Loop is: "To my sister Laura, who can understand, and to our sister Molly, who cannot."[49] Hofstadter explains in the preface that his younger sister Molly never developed the ability to speak or understand language.[50]

As a consequence of his attitudes about consciousness and empathy, Hofstadter has been a vegan for roughly half his life.[51]

In the 1982 novel 2010: Odyssey Two, Arthur C. Clarke's first sequel to 2001: A Space Odyssey, HAL 9000 is described by Dr. Chandra as being caught in a "Hofstadter–Möbius loop". The movie uses the term "H. Möbius loop".

On April 3, 1995, Hofstadter's book Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought was the first book ever sold by Amazon.com.[52]

Published works

Books

The books published by Hofstadter are (the ISBNs refer to paperback editions, where available):

  • Gödel, Escher, Bach: an Eternal Golden Braid (ISBN 0-465-02656-7) (1979)
  • Metamagical Themas (ISBN 0-465-04566-9) (collection of Scientific American columns and other essays, all with postscripts)
  • Ambigrammi: un microcosmo ideale per lo studio della creatività (ISBN 88-7757-006-7) (in Italian only)
  • Fluid Concepts and Creative Analogies (co-authored with several of Hofstadter's graduate students) (ISBN 0-465-02475-0)
  • Rhapsody on a Theme by Clement Marot (ISBN 0-910153-11-6) (1995, published 1996; volume 16 of series The Grace A. Tanner Lecture in Human Values)
  • Le Ton beau de Marot: In Praise of the Music of Language (ISBN 0-465-08645-4)
  • I Am a Strange Loop (ISBN 0-465-03078-5) (2007)
  • Surfaces and Essences: Analogy as the Fuel and Fire of Thinking, co-authored with Emmanuel Sander (ISBN 0-465-01847-5) (first published in French as L'Analogie. Cœur de la pensée; published in English in the U.S. in April 2013)

Papers

Hofstadter has written, among many others, the following papers:

Hofstadter has also written over 50 papers that were published through the Center for Research on Concepts and Cognition.[53]

Involvement in other books

Hofstadter has written forewords for or edited the following books:

Translations
  • Eugene Onegin: A Novel Versification from the Russian original of Alexander Pushkin), 1999. (ISBN 0-465-02094-1)
  • The Discovery of Dawn from the Italian original of Walter Veltroni, 2007. (ISBN 978-0-8478-3109-8)
  • That Mad Ache, co-bound with Translator, Trader: An Essay on the Pleasantly Pervasive Paradoxes of Translation from the French original of Francoise Sagan), 2009. (ISBN 978-0-465-01098-1)

See also

Notes
  1. GEB won the 1980 award for hardcover science.

References
  1. Hofstadter, Douglas R. (2008) [2003]. I Am a Strange Loop. New York, NY: Basic Books. ISBN 978-0-465-03079-8.
  2. "Golden Plate Awardees of the American Academy of Achievement". www.achievement.org. American Academy of Achievement.
  3. Hofstadter, Douglas Richard (1974). The Energy Levels of Bloch Electrons in a Magnetic Field (PhD thesis). University of Oregon. ProQuest 288009604.
  4. Hofstadter, D. R. (1982). "Who shoves whom around inside the careenium? Or what is the meaning of the word ?I??". Synthese. 53 (#2): 189–218. doi:10.1007/BF00484897. S2CID 46972278.
  5. "General Nonfiction" Archived February 26, 2012, at the Wayback Machine. Past winners and finalists by category. The Pulitzer Prizes. Retrieved March 17, 2012.
  6. A bedside book of paradoxes Archived March 26, 2017, at the Wayback Machine, New York Times
  7. "National Book Awards – 1980" Archived August 13, 2014, at the Wayback Machine. National Book Foundation. Retrieved March 7, 2012.
  8. Book Prizes – Los Angeles Times Festival of Books» Winners By Award Archived April 5, 2013, at the Wayback Machine. Events.latimes.com (November 22, 1963). Retrieved on 2013-10-06.
  9. Douglas Hofstadter at DBLP Bibliography Server
  10. Douglas Hofstadter's publications indexed by the Scopus bibliographic database. (subscription required)
  11. Stanford News Service,Nancy Hofstadter, widow of Nobel laureate in physics, dead at 87 Archived March 24, 2012, at the Wayback Machine, August 17, 2007.
  12. Hofstadter, Douglas (1976). "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields". Physical Review B. 14 (#6): 2239–2249. Bibcode:1976PhRvB..14.2239H. doi:10.1103/PhysRevB.14.2239.
  13. "Center for Research on Concepts and Cognition: Indiana University Bloomington". Archived from the original on April 11, 2018. Retrieved April 30, 2016.
  14. IU pages as faculty Archived December 31, 2003, at the Wayback Machine, IU distinguished faculty Archived February 25, 2004, at the Wayback Machine (see this announcement Archived December 16, 2007, at the Wayback Machine on March 21, 2007 speaker Archived December 16, 2007, at the Wayback Machine
  15. A Day in the Life of ... Douglas Hofstadter Archived December 30, 2007, at the Wayback Machine 2004
  16. Seminar: AI: Hope and Hype Archived June 6, 2007, at the Wayback Machine 1999
  17. Shore, Lys Ann (1988). "New Light on the New Age CSICOP's Chicago conference was the first to critically evaluate the New Age movement". The Skeptical Inquirer. 13 (#3): 226–235.
  18. "American Academy of Arts & Sciences". Archived from the original on July 28, 2012. Retrieved April 30, 2016.
  19. "Home - American Philosophical Society". Archived from the original on April 29, 2016. Retrieved April 30, 2016.
  20. Center for Research on Concepts and Cognition: Indiana University Bloomington Archived June 26, 1997, at the Wayback Machine. Cogsci.indiana.edu. Retrieved on October 6, 2013.
  21. An overview of Metacat Archived August 18, 2007, at the Wayback Machine 2003
  22. By Analogy: A talk with the most remarkable researcher in artificial intelligence today, Douglas Hofstadter, the author of Gödel, Escher, Bach Archived December 9, 2013, at the Wayback Machine Wired Magazine, November 1995
  23. Analogy as the Core of Cognition Archived April 11, 2008, at the Wayback Machine Review of Stanford lecture, February 2, 2006
  24. Center for Research on Concepts and Cognition Archived May 26, 2010, at the Wayback Machine
  25. Sounds like Bach Archived October 13, 2007, at the Wayback Machine
  26. Hofstadter, Douglas, To Err is Human; to Study Error-making is Cognitive Science. Together with David Moser. Michigan Quarterly Review, Vol. XXVIII, No. 2, 1989, pp. 185–215.
  27. Consciousness In The Cosmos: Perspective of Mind: Douglas Hofstadter Archived August 4, 2008, at the Wayback Machine
  28. Hofstadter, Douglas R. Le Ton Beau de Marot. New York: Basic Books, 1997, pp. 16–17.
  29. Hofstadter, Douglas R. Le Ton Beau de Marot, Chapter "How Jolly the Lot of an Oligoglot", New York: Basic Books, 1997, pp. 15–62.
  30. "People at the CRCC". The Center for Research on Concepts and Cognition. Archived from the original on February 22, 2014. Retrieved February 18, 2014.
  31. "Letter Spirit (part one): Emergent High-Level Perception of Gridletters Using Fluid Concepts". GARY MCGRAW, PH.D. Retrieved April 26, 2020.
  32. "Me, My Soul, and I". Wired. March 2007. Archived from the original on September 29, 2007. Retrieved December 10, 2007.
  33. The Mind Reader Archived March 1, 2017, at the Wayback Machine, New York Times Magazine, April 1, 2007
  34. Mean Chess-Playing Computer Tears at Meaning of Thought Archived March 17, 2015, at Wikiwix by Bruce Weber, February 19, 1996, New York Times
  35. Hofstadter, Douglas (1985). Metamagical Themas (PDF). p. 9. Archived from the original (PDF) on August 12, 2018. Retrieved December 19, 2018.
  36. "Will Spiritual Robots Replace Humanity By 2100?", April 1, 2000 Note: as of 2007, videos seem to be missing.
  37. "Moore's Law, Artificial Evolution, and the Fate of Humanity." In L. Booker, S. Forrest, et al. (eds.), Perspectives on Adaptation in Natural and Artificial Systems. New York: Oxford University Press, 2003.
  38. The Singularity Summit at Stanford Archived October 18, 2007, at the Wayback Machine 2006
  39. Trying to Muse Rationally about the Singularity Scenario Archived March 30, 2008, at the Wayback Machine 35 minute video, May 13, 2006
  40. Quotes from his 2006 Singularity Summit presentation Archived December 28, 2007, at the Wayback Machine
  41. "Staring EMI Straight in the Eye—and Doing My Best Not to Flinch." In David Cope, Virtual Music: Computer Synthesis of Musical Style, Cambridge, MA: The MIT Press, 2001.
  42. Victim of the Brain Archived August 17, 2007, at the Wayback Machine – 1988 docudrama about the ideas of Douglas Hofstadter
  43. Online implementation of his Reviews of this Book idea Archived January 7, 2009, at the Wayback Machine
  44. A Person Paper on Purity in Language Archived May 16, 2015, at the Wayback Machine by William Satire (alias Douglas R. Hofstadter), 1985 – a satirical piece, on the subject of sexist language
  45. Metamagical Themas, Douglas R. Hofstadter, Basic Books, New Yoork (1985), see preface, introduction, contents listing.
  46. French and Italian Archived December 12, 2007, at the Wayback Machine Spring 1996, Vol. X
  47. https://www.hoosiertimes.com/search/?q=hofstadter+baofen
  48. Piano Music by Douglas Hofstadter (audio CD), 2000, ISBN 1-57677-143-1
  49. Hofstadter, Douglas R. I Am a Strange Loop, p. v. Basic Books, 2007.
  50. Hofstadter, Douglas R. I Am a Strange Loop, p. xi. Basic Books, 2007. "No one knew what it was, but Molly wasn't able to understand language or to speak (nor is she to this day, and we never did find out why)."
  51. Gardner, Martin (August 2007). "Do Loops Explain Consciousness? Review of I Am a Strange Loop" (PDF). Notices of the American Mathematical Society. 54 (#7): 853. Archived (PDF) from the original on February 16, 2008. Retrieved December 10, 2007.
  52. 2012
  53. "Archived copy". Archived from the original on June 20, 2004. Retrieved May 27, 2004.CS1 maint: archived copy as title (link) CRCC Publications offline
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