I Am a Strange Loop
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Page 126 a trio of mathematicians… These are Yann Bugeaud, Maurice Mignotte, and Samir Siksek. It turns out that to prove that 144 is the only square in the Fibonacci sequence (other than 1) does not require highly abstract ideas, although it is still quite subtle. This was accomplished in 1964 by John H. E. Cohn.
Page 128 Gödel’s analogy was very tight… The essence and the meaning of Gödel’s work are well presented in many books, including [Nagel and Newman], [DeLong], [Smullyan 1961], [Jeffrey], [Boolos and Jeffrey], [Goodstein], [Goldstein], [Smullyan 1978], [Smullyan 1992], [Wilder], [Kneebone], [Wolf], [Shanker], and [Hofstadter 1979].
Page 129 developed piecemeal over many centuries… See [Nagel and Newman], [Wilder], [Kneebone], [Wolf ], [DeLong], [Goodstein], [Jeffrey], and [Boolos and Jeffrey].
Page 135 Anything you can do, I can do better!… My dear friend Dan Dennett once wrote (in a lovely book review of [Hofstadter and FARG], reprinted in [Dennett 1998]) the following sentence: “‘Anything you can do I can do meta’ is one of Doug’s mottoes, and of course he applies it, recursively, to everything he does.”
Well, Dan’s droll sentence gives the impression that Doug himself came up with this “motto” and actually went around saying it (for why else would Dan have put it in quote marks?). In fact, I had never said any such thing nor thought any such thought, and Dan was just “going me one meta”, in his own inimitable way. To my surprise, though, this “motto” started making the rounds and people quoted it back to me as if I really had thought it up and really believed it. I soon got tired of this because, although Dan’s motto is clever and funny, it does not match my self-image. In any case, this note is just my little attempt to squelch the rumor that the above-displayed motto is a genuine Hofstadter sentence, although I suspect my attempt will not have much effect.
Page 137 suppose you wanted to know if statement X is true or false… The dream of a mechanical method for reliably placing statements into two bins — ‘true’ and ‘false’ — is known as the quest for a decision procedure. The absolute nonexistence of a decision procedure for truth (or for provability) is discussed in [DeLong], [Boolos and Jeffrey], [Jeffrey], [Hennie], [Davis 1965], [Wolf], and [Hofstadter 1979].
Page 139 No formula can literally contain… [Nagel and Newman] presents this idea very clearly, as does [Smullyan 1961]. See also [Hofstadter 1982].
Page 139 an elegant linguistic analogy… See [Quine] for the original idea (which is actually a variation of Gödel’s idea (which is itself a variation of an idea of Jules Richard (which is a variation of an idea of Georg Cantor (which is a variation of an idea of Euclid (with help from Epimenides))))), and [Hofstadter 1979] for a variation on Quine’s theme.
Page 147 “…and Related Systems (I)”… Gödel put a roman numeral at the end of the title of his article because he feared he had not spelled out sufficiently clearly some of his ideas, and expected he would have to produce a sequel. However, his paper quickly received high praise from John von Neumann and other respected figures, catapulting the unknown Gödel to a position of great fame in a short time, even though it took most of the mathematical community decades to absorb the meaning of his results.
Page 150 respect for …the most mundane of analogies… See [Hofstadter 2001] and [Sander], as well as Chapter 24 in [Hofstadter 1985] and [Hofstadter and FARG].
Page 159 X’s play is so mega-inconsistent… This should be heard as “X’s play is omega-inconsistent”, which makes a phonetic hat-tip to the metamathematical concepts of omega-inconsistency and omega-incompleteness, discussed in many books in the Bibliography, such as [DeLong], [Nagel and Newman], [Hofstadter 1979], [Smullyan 1992], [Boolos and Jeffrey], and others. For our more modest purposes here, however, it suffices to know that this “o”-containing quip, plus the one two lines below it, is a play on words.
Page 160 Indeed, some years after Gödel, such self-affirming formulas were concocted… See [Smullyan 1992], [Boolos and Jeffrey], and [Wolf].
Page 164 Why would logicians …give such good odds… See [Kneebone], [Wilder], and [Nagel and Newman], for reasons to believe strongly in the consistency of PM-like systems.
Page 165 not only although…but worse, because… For another treatment of the perverse theme of “although” turning into “because”, see Chapter 13 of [Hofstadter 1985].
Page 166 the same Gödelian trap would succeed in catching it… For an amusing interpretation of the infinite repeatability of Gödel’s construction as demonstrating the impossibility of artificial intelligence, see the chapter by J. R. Lucas in [Anderson], which is carefully analyzed (and hopefully refuted) in [DeLong], [Webb], and [Hofstadter 1979].
Page 167 called “the Hilbert Program”… See [DeLong], [Wolf ], [Kneebone], and [Wilder].
Page 170 In that most delightful though most unlikely of scenarios… [DeLong], [Goodstein], and [Chaitin] discuss non-Gödelian formulas that are undecidable for Gödelian reasons.
Page 172 No reliable prim/saucy distinguisher can exist… See [DeLong], [Boolos and Jeffrey] , [Jeffrey], [Goodstein], [Hennie], [Wolf ], and [Hofstadter 1979] for discussions of many limitative results such as this one (which is Church’s theorem).
Page 172 It was logician Alfred Tarski who put one of the last nails… See [Smullyan 1992] and [Hofstadter 1979] for discussions of Tarski’s deep result. In the latter, there is a novel approach to the classical liar paradox (“This sentence is not true”) using Tarski’s ideas, with the substrate taken to be the human brain instead of an axiomatic system.
Page 172 what appears to be a kind of upside-down causality… See [Andersen] for a detailed technical discussion of downward causality. Less technical discussions are found in [Pattee] and [Simon]. See also Chapters 11 and 20 in [Hofstadter and Dennett], and especially the Reflections. [Laughlin] gives fascinating arguments for the thesis that in physics, the macroscopic arena is more fundamental or “deeper” than the microscopic.
Page 174 leaving just a high-level picture of information-manipulating processes… See [Monod], [Berg and Singer], [Judson], and Chapter 27 of [Hofstadter 1985].
Page 177 symbols in our respective brains… See [Hofstadter 1979], especially the dialogue “Prelude… Ant Fugue” and Chapters 11 and 12, for a careful discussion of this notion.
Page 178 the forbidding and inaccessible level of quarks and gluons… See [Weinberg 1992] and [Pais 1986] for attempts at explanations of these incredibly abstruse notions.
Page 178 the only slightly more accessible level of genes… See [Monod], [Berg and Singer], [Judson], and Chapter 27 (“The Genetic Code: Arbitrary?”) in [Hofstadter 1985].
Page 179 we…best understand our own actions as… See [Dennett 1987] and [Dennett 1998].
Page 181 embellished by a fantastic folio of alternative versions… [Steiner 1975] has a rich and provocative discussion of “alternity”, and the dialogue “Contrafactus” in [Hofstadter 1979] features an amusing scenario involving “subjunctive instant replays”. See also [Kahneman and Miller] and Chapter 12 of [Hofstadter 1985] for further musings on the incessantly flickering presence of counterfactuals in the subconscious human mind. [Hofstadter and FARG] describes a family of computational models of human thought processes in which making constant forays into alternity is a key architectural feature.
Page 182 housing a loop of self-representation… See [Morden], [Kent], and [Metzinger].
Page 186 as the years pass, the “I” converges and stabilizes itself… See [Dennett 1992].
Page 188 we cannot help attributing reality to our “I” and to those of other people… See [Kent], [Dennett 1992], [Brinck], [Metzinger], [Perry], and [Hofstadter and Dennett].
Page 189 I was most impressed when I read about “Stanley”, a robot vehicle… See [Davis 2006].
Page 193 just a big spongy bulb of inanimate molecules… I suppose almost any book on the brain will convince one of this, but [Penfield and Roberts] did it to me as a teen-ager.
Page 194 pioneering roboticist and provocative writer
Hans Moravec… For some of Moravec’s more provocative speculations about the near-term future of humanity, see [Moravec].
Page 194 from the organic chemistry of carbon… See Chapter 22 in [Hofstadter and Dennett], in which John Searle talks about “the right stuff”, which underwrites what he terms “the semantic causal powers of the brain”, a rather nice-sounding but murky term by which Searle means that when a human brain, such as his own or, say, that of poet Dylan Thomas, makes its owner come out with words, those words don’t just seem to stand for something, they really do stand for something. Unfortunately, in the case of poet Thomas, most of his output, though it sounds rather nice, is so full of murk that one has to wonder what sort of “stuff” could possibly make up the brain behind it.
Page 199 its symbol-count might well exceed “Graham’s constant”… See [Wells 1986].
Page 208 For those who enjoy the taboo thrills of non-wellfounded sets… See [Barwise and Moss].
Page 209 the deeper and richer an organism’s categorization equipment is… See [Hofstadter 2001].
Page 233 a devilishly clever bon mot by David Moser… One evening not long after we were married, Carol and I invited some friends over for an Indian dinner at our house in Ann Arbor. Melanie Mitchell and David Moser, well aware of Carol’s terrific Indian cooking, were delighted to come. It turned out, however, that at the last minute, our oldest guests, in their eighties, called up to tell us that they couldn’t handle very spicy foods, which unfortunately torpedoed Carol’s cooking plans. Somehow, though, she turned around on a dime and prepared a completely different yet truly delicious repast. A couple of hours after dinner was over, after a very lively discussion, most of our guests took off, leaving just David, Melanie, Carol, and me. We chatted on for a while, and finally, as they were about to hit the road, Carol casually reminded them of what she had originally intended to fix and told them why she hadn’t been able to follow through on her promise. Quick as a wink, David, feigning great indignation, burst out, “Why, you Indian-dinner givers, you!”
Page 233 her personal gemma (to borrow Stanislaw Lem’s term …)… See “Non Serviam” in [Hofstadter and Dennett], which is a virtuosic philosophical fantasy masquerading as a book review (of a book that, needless to say, is merely a figment of Lem’s imagination).
Page 239 someone trying to grapple with quantum-mechanical reality… [Pais 1986], [Pais 1992], and [Pullman] portray the transition period between the Bohr atom and quantum mechanics, while [Jauch] and [Greenstein and Zajonc] chart remaining mysteries.
Page 239 it might be tempting for some readers to conclude that in the wake of Carol’s death… See Chapter 15 of [Hofstadter 1997], another place where I discuss many of these ideas.
Page 242 meaning of the term “universal machine”… See [Hennie] and [Boolos and Jeffrey].
Page 248 concepts are active symbols in a brain… See Chapter 11 of [Hofstadter 1979].
Page 252 a marvelous pen-and-ink “parquet deformation” drawn in 1964… For a dozen-plus examples of this subtle Escher-inspired art form, see Chapter 10 of [Hofstadter 1985].
Page 260 It is not easy to find a strong, vivid metaphor to put up against the caged-bird metaphor… The idea of a soul distributed over many brains brought to my mind an image from solid-state physics, the field in which I did my doctoral work. A solid is a crystal, meaning a periodic lattice of atoms in space, like the trees in an orchard but in three dimensions instead of two. In some solids (those that do not conduct electricity), the electrons “hovering” around each atomic nucleus are so tightly bound that they never stray far from that nucleus. They are like butterflies that hover around just one tree in the orchard, never daring to venture as far as the next tree. In metals, by contrast, which are excellent conductors, the electrons are not timid stay-at-homes stuck to one tree, but boldly float around the entire lattice. This is why metals conduct so well.
Actually, the proper image of an electron in a metal is not that of a butterfly fickly fluttering from one tree to another, never caring where it winds up, but of an intensity pattern distributed over the entire crystal at once — in some places more intense, in other places less so, and changing over time. One electron might better be likened to an entire swarm of orange butterflies, another electron to a swarm of red butterflies, another to a swarm of blue butterflies, and so on, with each swarm spread about the whole orchard, intermingling with all the others. Electrons in metals, in short, are anything but tightly bound dots; they are floating patterns without any home at all.
But let’s not lose track of the purpose of all this imagery, which is to suggest helpful ways of imagining what a human soul’s essence is. If we map each tree (or nucleus) in the crystal lattice onto a particular human brain, then in the tight-binding model (which corresponds to the caged-bird metaphor), each brain would possess a unique soul, represented by the cloud of timid butterflies that hover around it and it alone. By contrast, if we think of a metal, then the cloud is spread out across the whole lattice — which is to say, shared equally among all the trees (or nuclei). No tree is privileged. In this image, then (which is close to Daniel Kolak’s view in I Am You), each human soul floats among all human brains, and its identity is determined not by its location but by the undulating global pattern it forms.
These are extremes, but nothing keeps us from imagining a halfway situation, with many localized swarms of butterflies, each swarm floating near a single tree but not limited to it. Thus a red swarm might be centered on tree A but blur out to the nearest dozen trees, and a blue swarm might be blurrily centered on tree B, a yellow swarm around tree C, etc. Each tree would be the center of just one swarm, and each swarm would have just one principal tree, but the swarms would interpenetrate so intimately that it would be hard to tell which swarm “belonged” to which tree, or vice versa.
This peculiar and surreal tale, launched in solid-state physics but winding up with imagery of interpenetrating swarms of colored butterflies fluttering in an orchard, gives as clear a picture as I can paint of how a human soul is spread among brains.
Page 264 Many of these ideas were explored …in his philosophical fantasy “Where Am I?”… This classic piece can be found in [Dennett 1978] and in [Hofstadter and Dennett].
Page 267 internal conflict between several “rival selves”… Chapter 13 of [Dennett 1991] gives a careful discussion of multiple personality disorder. See also [Thigpen and Cleckley], from which a famous movie was made. See also [Minsky 1986] and Chapter 33 of [Hofstadter 1985] for views of a normal self as containing many competing subselves.
Page 267 in such cases Newtonian physics goes awry… See [Hoffmann] for a discussion of the subtle relationship between relativistic and Newtonian physics.
Page 271 every entity…is conscious… See [Rucker] for a positive view of panpsychism.
Page 276 because now they want the symbols themselves to be perceived… See the careful debunking in [Dennett 1991] of what its author terms the “Cartesian Theater”.
Page 277 to trigger just one familiar pre-existing symbol… This sentence is especially applicable to the nightmare of preparing an index. Only if one has slaved away for weeks on a careful index can one have an understanding of how grueling (and absurd) the task is.
Page 278 when its crust is discarded and its core is distilled… See [Sander], [Kahneman and Miller], [Kanerva], [Schank], [Boden], and [Gentner et al.] for discussions of the analogy-based mechanisms of memory retrieval, which underlie all human cognition.
Page 279 to simplify while not letting essence slip away… See [Hofstadter 2001], [Sander], and [Hofstadter and FARG]. To figure out how to give a computer the rudiments of this ability has been the Holy Grail of my research group for three decades now.
Page 279 There is not some special “consciousness locus”… See [Dennett 1991].
Page 282 but we are getting ever closer… See [Monod], [Cordeschi], and [Dupuy 2000] for clear discussions of the emergence of goal-orientedness (i.e., teleology) from
feedback.
Page 283 a physical vortex, like a hurricane or a whirlpool… See Chapter 22 of [Hofstadter 1985] for a discussion of the abstract essence of hurricanes.
Page 283 every integer is the sum of at most four squares… See [Hardy and Wright] and [Niven and Zuckerman] for this classic theorem, the simplest case of Waring’s theorem.
Page 285 to see that brilliant purple color of the flower… See [Chalmers] for a spirited defense of the notion of qualia, and see [Dennett 1991], [Dennett 1998], [Dennett 2005], and [Hofstadter and Dennett], which do their best to throw a wet blanket on the idea.
Page 287 There is no meaning to the letter “b”… See the dialogue “Prelude… Ant Fugue”
(found in both [Hofstadter 1979] and [Hofstadter and Dennett]) for a discussion of how meanings at a high level can emerge from meaningless symbols at a low level.
Page 293 the notion that consciousness is a novel kind of quantum phenomenon… See [Penrose], which views consciousness as an intrinsically quantum-mechanical phenomenon, and [Rucker], which views consciousness as uniformly pervading everything in the universe.
Page 295 Taoism and Zen long ago sensed this paradoxical state… Far and away the best book I have read on these spiritual approaches to life is [Smullyan 1977], but [Smullyan 1978] and [Smullyan 1983] also contain excellent pieces on the topic. These ideas are also discussed in Chapter 9 of [Hofstadter 1979], but from a skeptical point of view.
Page 296 the story of an “I” is a tale about a central essence… See [Dennett 1992] and [Kent].
Page 298 The…self-pointing loop that the pronoun “I” involves … See [Brinck] and [Kent].
Page 299 This is what John von Neumann unwittingly revealed… See [von Neumann] for a very difficult and [Poundstone] for a very lucid discussion of self-replicating automata.
See Chapters 2 and 3 of [Hofstadter 1985] for a simpler discussion of the same ideas. Chapter 16 of [Hofstadter 1979] carefully spells out the mapping between Gödel’s self-referential construction and the self-replicating mechanisms at the core of life.