I Am a Strange Loop

NOTES

Page xi gave me the impetus to read a couple of lay-level books about the human brain… These were [Pfeiffer] and [Penfield and Roberts]. Another early key influence was [Wooldridge].

Page xi the physical basis…of being…an “I”, which… Placing commas and periods outside quotation marks when they are not part of what is being quoted exhibits greater logic than does American usage, which puts them inside regardless of circumstance. In this book, the logical convention (also the standard in British English) is adopted.

Page xiv Hofstadter’s Law… This comes from Chapter V of [Hofstadter 1979].

Page xiv “What is it like to be a bat?”… See Chapter 24 in [Hofstadter and Dennett].

Page xv I have spent nearly thirty years… See, for instance, [Hofstadter and Moser], [Hofstadter and FARG], [Hofstadter 1997], and [Hofstadter 2001].

Page xviii virtually every thought in this book…is an analogy… See [Hofstadter 2001].

Page xviii not indulging in Pushkinian digressions… See James Falen’s sparkling anglicization of Pushkin’s classic novel-in-verse Eugene Onegin [Pushkin 1995], or see my own translation [Pushkin 1999]. There is no sublimer marriage of form to content than Eugene Onegin.

Page xviii typeset it down to the finest level of detail… In this book, one of my chief esthetic concerns has been where page breaks fall. A cardinal rule has been that no paragraph (or section) should ever break in such a way that only one line of it occurs at the top or bottom of a page. Another guiding principle has been that the interword spacing in each line should look pleasing, and, in particular, not too loose (which is a frequent and annoying eyesore in computer-set text). In order to avoid such blemishes, I have done touch-up rewriting, often quite extensive, of just about every paragraph in the book. Page xviii itself is a typical example of the end result. And of course the page you are right now reading (and that I am right now touching up so that it will please your eye) is another such example.

The foregoing esthetic constraints (along with a number of others that I won’t describe here) amount to random darts being thrown at every page in the book, with each dart saying to me, in effect, “Here — don’t you think you could rewrite this sentence so that it not only looks better but also makes its point even more clearly and elegantly?” Some authors might find this tiresome, but I freely confess that I love these random darts and the two-sided challenges that they offer me, and I have worked extremely hard to meet those challenges throughout. There is not a shadow of a doubt that form–content pressures — relentless, intense, and unpredictable — have greatly improved the quality of this book, not only visually but also intellectually.

For a more explicit spelling-out of my views on the magical power of form–content interplay, see [Hofstadter 1997], especially its Introduction and Chapter 5.

Page 5 no machine can know what words are, or mean… This ancient idea is the rallying cry of many philosophers, such as John Searle. See Chapter 20 of [Hofstadter and Dennett].

Page 5 the laws of whose operation are arithmetical… This is an allusion to the idea that a “Giant Electronic Brain”, whose very fiber is arithmetical, could act indistinguishably from a human or animal brain by modeling the arithmetical behavior of all of its neurons. This would give rise to a kind of artificial intelligence, but very different from models in which the basic entities are words or concepts governed by rules that reflect the abstract flow of ideas in a mind rather than the microscopic flow of currents and chemicals in biological hardware. Chapter XVII of [Hofstadter 1979], Chapter 26 of [Hofstadter and Dennett], and Chapter 26 of [Hofstadter 1985] all represent elaborations of this subtle distinction, which I was beginning to explore in my teens.

Page 10 I don’t know what effect it had on her feelings about the picture… With some trepidation, I recently read aloud this opening section of my book to my mother, who, at almost 87, can only move around her old Stanford house in a wheelchair, but who remains sharp as a tack and intensely interested in the world around her. She listened with care and then remarked, “I must have changed a lot since then, because now, those pictures mean everything to me. I couldn’t live without them.” I doubt that what I said to her that gloomy day nearly sixteen years ago played much of a role in this evolution of her feelings, but I was glad in any case to hear that she had come to feel that way.

Page 10 a tomato is a desireless, soulless, nonconscious entity… On the other hand, [Rucker] proposes that tomatoes, potatoes, cabbages, quarks, and sealing-wax are all conscious.

Page 11 a short story called “Pig”… Found in [Dahl].

Page 16 In his preface to the volume of Chopin’s études… All the prefaces that Huneker wrote in the Schirmer editions can be found in [Huneker].

Page 18 What gives us word-users the right to make… See [Singer and Mason].

Page 20 it is made of ‘the wrong stuff’… That brains but not computers are made of “the right stuff” is a slogan of John Searle. See Chapter 20 in [Hofstadter and Dennett].

Page 23 Philosophers of mind often use the terms… See, for example, [Dennett 1987].

Page 25 “What do I mean…by ‘brain research’?”… See [Churchland], [Dennett 1978], [Damasio], [Flanagan], [Hart], [Harth], [Penfield], [Pfeiffer], and [Sperry].

Page 26 these are all legitimate and important objects of neurological study… See [Damasio], [Kuffler and Nicholls], [Wooldridge], and [Penfield and Roberts].

Page 26 abstractions are central…in the study of the brain… See [Treisman], [Minsky 1986], [Schank], [Hofstadter and FARG], [Kanerva], [Fauconnier], [Dawkins], [Blackmore], and [Wheelis] for spellings-out of these abstract ideas.

Page 27 Just as the notion of “gene” as an invisible entity that enabled… See [ Judson].

Page 27 and just as the notion of “atoms” as the building blocks… See [Pais 1986], [Pais 1991], [Hoffmann], and [Pullman].

Page 28 Turing machines are…idealized computers… See [Hennie] and [Boolos and Jeffrey].

Page 29 In his vivid writings, Searle gives… See Chapter 22 of [Hofstadter and Dennett].

Page 29 one particular can that would “pop up”… In his smugly dismissive review [Searle] of [Hofstadter and Dennett], Searle states: “So let us imagine our thirst-simulating program running on a computer made entirely of old beer cans, millions (or billions) of old beer cans that are rigged up to levers and powered by windmills. We can imagine that the program simulates the neuron firings at the synapses by having beer cans bang into each other, thus achieving a strict correspondence between neuron firings and beer-can bangings. And at the end of the sequence a beer can pops up on which is written ‘I am thirsty.’ Now, to repeat the question, does anyone suppose that this Rube Goldberg apparatus is literally thirsty in the sense in which you and I are?”

Page 30 Dealing with brains as multi-level systems… See [Simon], [Pattee], [Atlan], [Dennett 1987], [Sperry], [Andersen], [Harth], [Holland 1995], [Holland 1997], and the dialogue “Prelude… Ant Fugue” in [Hofstadter 1979] or in [Hofstadter and Dennett].

Page 31 such as a column in the cerebral cortex… See [Kuffler and Nicholls].

Page 31 I once saw a book whose title was “Molecular Gods…” This was [Applewhite].

Page 31 to quote here a short passage from Sperry’s essay… Taken from [Sperry].

Page 32 taken from “The Floor”… See [Edson], which is a thin, remarkably vivid, highly surrealistic, often hilarious, and yet profoundly depressing collection of prose poems.

Page 33 such macroscopic phenomena as friction… A beautiful and accessible account of the emergence of everyday phenomena (such as how paper tears) out of the surrealistically weird quantum-mechanical substrate of our world is given in [Chandrasekhar].

Page 34 quarks, gluons, W and Z bosons… See [Pais 1986] and [Weinberg 1992].

Page 35 Drastic simplification is what allows us to…discover abstract essences… See [Kanerva], [Kahneman and Miller], [Margolis], [Sander], [Schank], [Hofstadter and FARG], [Minsky 1986], and [Gentner et al.].

Page 38 641, say… I chose the oddball integer 641 because it plays a famous role in the history of mathematics. Fermat conjectured that all integers of the form are prime, but Euler discovered that 641 (itself a prime) divides , thus refuting Fermat’s conjecture. See [Wells 1986], [Wells 2005], and [Hardy and Wright].

Page 41 Deep understanding of causality… See [Pattee], [Holland 1995], [Holland 1997], [Andersen], [Simon], and Chapter 26 of [Hofstadter 1985].

Page 45 The Careenium… Chapter 25 of [Hofstadter 1985] is a lengthy Achilles–Tortoise dialogue spelling out the careenium metaphor in detail.

Page 49 The effect…was explained…by Albert Einstein… See [Hoffmann] and [Pais 1986].

Page 49 From this perspective, there are no simmballs, no symbols… This view approaches the extreme reductionist philosophy expressed in [Unger 1979] and also in [Unger 1979].

Page 52 Why does this move to a goal-oriented — that is, teleological — shorthand… See [Monod], [Cordeschi], [Haugeland 1981], and [Dupuy 2000].

Page 53 In the video called “Virtual Creatures” by Karl Sims… This is found easily on the Web.

Page 53 a strong pressure to shift …to the goal-oriented level of cybernetics… See [Dupuy 2000], [Monod], [Cordeschi], [Simon], [Andersen], and Chapter 11 in [Hofstadter and Dennett], which discusses a trio of related “isms” — holism, goalism, and soulism.

Page 54 the story of a sultan who commanded… Found in the charming old book [Gamow].

Page 55 contains the seeds of its own destruction… Compare this scenario of self-breaking to the story recounted in the dialogue “Contracrostipunctus” in [Hofstadter 1979].

Page 57 I stumbled upon …a little paperback… Of course this was [Nagel and Newman].

Page 57 I’m sure I didn’t think “he or she”… See Chapters 7 and 8 of [Hofstadter 1985].

Page 60 pushed my luck and invented the more threeful phrase… Although I didn’t know it, I was dimly sensing the infinite hierarchy of arithmetical operations and what I would later come to know as “Ackermann’s function”. See [Boolos and Jeffrey] and [Hennie].

Page 61 a pathological retreat from common sense… I cannot resist pointing out that Principia Mathematica opens with a grand flourish of self-reference, its first sentence unabashedly declaring: “The mathematical treatment of the principles of mathematics, which is the subject of the present work, has arisen from the conjunction of two different studies, both in the main very modern.” Principia Mathematica thus points at itself through the proud phrase “the present work” — exactly the kind of self-pointer that, in more formal contexts, its authors were at such pains to forbid categorically. Perhaps more weirdly, the chapter in which the self-reference–banning theory of types is presented also opens self-referentially: “The theory of logical types, to be explained in the present Chapter, recommended itself to us in the first instance by its ability to solve certain contradictions…” Note finally that the pronoun “us” is yet another self-pointer that Russell and Whitehead have no qualms using. Were they not aware of these ironies?

Page 62 the topic of self-reference in language… See Chapters 1–4 of [Hofstadter 1985].

Page 62 This pangram tallies… This perfectly self-tallying or self-inventorying “pangram” was discovered by Lee Sallows using an elaborate analog computer that he built.

I have often mused about a large community of sentences somewhat like Sallows’, each one inventorying not only itself (i.e., giving 26 letter-counts as above), but in addition some or perhaps all of the others. Thus each sentence would be far, far longer than Sallows’ pangram. However, in my fantasy, these “individuals”, unlike Sallows’ remarkable sentence, do not all give accurate reports. Some of what they say is dead wrong. In the self-inventorying department, I imagine most of them as being fairly accurate (most of their 26 “first-person” counts would be precisely right, with just a few perhaps being a little bit off). On the other hand, each sentence’s inventory of other sentences would vary in accuracy, from being somewhat close to being wildly far off.

Needless to say, this is a metaphor for a society of interacting human beings, each of whom has a fairly accurate self-image and less accurate images of others, often based on very quick and inaccurate glances. Two sentences that “know each other well” (i.e., that have reasonably accurate though imperfect inventories of each other) would be the analogue of good friends, whereas two sentences that have rough, partial, or vacuous representations of each other would be the analogue of strangers.

A more complex variation on this theme involves a population of Sallows-type sentences varying in time. At the outset, they would all be filled with random numbers, but then they would all get updated in parallel. Specifically, each one would replace its wrong inventories by counting letters inside itself and in a few other sentences, and replacing the wrong values by the values just found. Of course, since everything is a moving target, the letter-counts would still be wrong, but hopefully over the course of a long series of such parallel iterations, each sentence would tend, at least on average, to gain greater accuracy, especially concerning itself, and simultaneously to form a small clique of “friends” (sentences that it inventories fairly fully and well), while remaining remote from most members of the population (i.e., representing them at best sparsely and with many errors, or perhaps not even at all). This is a kind of caricature of my ideas about people “living inside each other”, proposed in Chapters 15 through 18.

Page 63 Perhaps there is no harm… Quoted from [Skinner] in George Brabner’s letter.

Page 63 I wrote a lengthy reply to it… This is found in Chapter 1 of [Hofstadter 1985].

Page 68 If dogs were a bit more like robots… As I was putting the finishing touches on these notes, my children and I flew out to California for Christmas break. We were gliding low, approaching the San Jose airport at night, when Danny, who was peering out the window, said to me, “You know what I just saw?” “What?” I replied, having not the foggiest idea. He said, “A parking lot packed with cars whose headlights and taillights were all flashing on and off at random!” “Why were they all doing that?” I asked, a bit densely. Danny instantly supplied the answer: “Their alarm systems were all triggering each other. I know that’s what it was, because I’ve seen fireworks set car alarms off.” Seeing this in my mind’s eye, I grinned from ear to ear with delight and amazement, all the more so since Danny hadn’t read any of my manuscript and had no idea how relevant his sighting of reverberant honking and flashing was to my book — in fact to the chapter that I was writing notes for just then (Chapter 5). Danny’s reverberant parking lot truly put reverberant barking to shame, and what an infernal racket it must have been for people down on the ground! And yet, as observed from above by chance voyeurs in the plane, it was a totally silent, surrealistic vision of robots who had gotten one another all excited, and who certainly weren’t about to calm down, as dogs will. What a stupendous last-minute addition to my book!

Page 69 the amazing visual universe discovered around 1980… See [Peitgen and Richter].

Page 76 winds up triggering a small set… See [Kanerva] and [Hofstadter and FARG].

Page 77 Suppose we begin with a humble mosquito… See [Griffin] and [Wynne]. The latter contains a remarkable account of analogy-making by bees, of all creatures!

Page 80 cars that drive themselves down …highways or across rocky deserts… See [Davis 2006].

Page 82 structure that represents itself (i.e., the dog itself, not the symbol itself !)… This sounds like a joke, but not entirely. When it comes to the self-symbols of humans — their “I” ’s — much of the structure of the “I” involves pointers that point right back at the abstraction “I”, and not just at the body. This is discussed in Chapters 13 and 16.

Page 83 their category systems became arbitrarily extensible… I defend this point of view in [Hofstadter 2001]. For more on human categories, see [Sander], [Margolis], [Minsky 1986], [Schank], [Aitchison], [Fauconnier], [Hofstadter 1997], and [Gentner et al.].

Page 85 memories of episodes can be triggered… See [Kanerva], [Schank], and [Sander].

Page 86 That deep and tangled self-model is what “I”-ness is all about… See [Dennett 1991], [Metzinger], [Horney 1942], [Horney 1945], [Wheelis], [Nørretranders], and [Kent].

Page 89 Abstraction piled on abstraction… Should anyone care to get a taste of this, try reading [Ash and Gross] all the way to the end. It’s a bit like ordering “Indian hot” in an authentic Indian restaurant — you’ll wonder why you ever did.

Page 91 radicals, such as Évariste Galois… The great Galois was indeed a young radical, which led to his absurdly tragic death in a duel on his twenty-first birthday, but the phrase “solution by radicals” really refers to the taking of nth roots, called “radicals”. For a shallow, a medium, and a deep dip into Galois’ immortal, radical insights into hidden mathematical structures, see [Livio], [Bewersdorff ], and [Stewart], respectively.

Page 95 there is a special type of abstract structure or pattern… “Real Patterns” in [Dennett 1998] argues powerfully for the reality of abstract patterns, based on John Conway’s cellular automaton known as the “Game of Life”. The Game of Life itself is presented ideally in [Gardner], and its relevance to biological life is spelled out in [Poundstone].

Page 102 I am sorry to say, now hackneyed… I have long loved Escher’s art, but as time has passed, I have found myself drawn ever more to his early non-paradoxical landscapes, in which I see hints everywhere of his sense of the magic residing in ordinary scenes. See [Hofstadter 2002], an article written for a celebration of Escher’s 100th birthday.

Page 103 Is there, then, any genuine strange loop — a paradoxical structure that… Three excellent books on paradoxes are [Falletta], [Hughes and Brecht], and [Casati and Varzi 2006].

Page 104 an Oxford librarian named G. G. Berry… Only two individuals are thanked by the (nearly) self-sufficient authors of Principia Mathematica, and G. G. Berry is one of them.

Page 108 Chaitin and others went on… See [Chaitin], packed with stunning, strange results.

Page 113 written in PM notation as… I have here borrowed Gödel’s simplified version of PM notation instead of taking the symbols directly from the horses’ mouths, for those would have been too hard to digest. (Look at page 123 and you’ll see what I mean.)

Page 114 the sum of two squares… See [Hardy and Wright] and [Niven and Zuckerman].

Page 114 the sum of two primes… See [Wells 2005], an exquisite garden of delights.

Page 116 The passionate quest after order in an apparent disorder is what lights their fires… See [Ulam], [Ash and Gross], [Wells 2005], [Gardner], [Bewersdorff ], and [Livio].

Page 117 Nothing happens “by accident” in the world of mathematics… See [Davies].

Page 118 Paul Erdös once made the droll remark… Erdös, a devout matheist, often spoke of proofs from “The Book”, an imagined tome containing God’s perfect proofs of all great truths. For my own vision of “matheism”, see Chapter 1 of [Hofstadter and FARG].

Page 119 Variations on a Theme by Euclid… See [Chaitin].

Page 120 God does not play dice… See [Hoffmann], one of the best books I have ever read.

Page 121 many textbooks of number theory prove this theorem… See [Hardy and Wright] and [Niven and Zuckerman].

Page 122 About a decade into the twentieth century… The history of the push to formalize mathematics and logic is well recounted in [DeLong], [Kneebone], and [Wilder].

Page 122 a young boy was growing up in the town of Brünn… See [Goldstein] and [Yourgrau].

Page 125 Fibonacci …explored what are now known as the “Fibonacci numbers”… See [Huntley].

Page 125 This almost-but-not-quite-circular fashion… See [Péter] and [Hennie].

Page 126 a vast team of mathematicians… A recent book that purports to convey the crux of the elusive ideas of this team is [Ash and Gross]. I admire their chutzpah in trying to communicate these ideas to a wide public, but I suspect it is an impossible task.

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.

Page 300 too marbelous for words… Borrowing a few words from a love song by Johnny Mercer and Richard Whiting, sung in an unsurpassable fashion by Frank Sinatra.

Page 300 with alacrity, celerity, assiduity, vim, vigor, vitality… My father’s friend Bob Herman (a top-notch physicist who famously co-predicted the cosmic background radiation fifteen years before it was observed) loved to recite this riddle, putting on a strong Yiddish accent: “A tramp in the woods happened upon a hornets’ nest. When they stung him with alacrity, celerity, assiduity, vim, vigor, vitality, savoir-faire, and undue velocity, ‘Oh!’, he mused, counting his bumps, ‘If I had as many bumps on the left side of my right adenoid as six and three-quarters times seven-eighths of those between the heel of Achilles and the circumference of Adam’s apple, how long would it take a boy rolling a hoop up a moving stairway going down to count the splinters on a boardwalk if a horse had six legs?’ ” And so I thought I’d give a little posthumous hat-tip to Bob.

Page 305 Dan calls such carefully crafted fables ‘intuition pumps’… Dennett introduced his term “intuition pump”, I believe, in the Reflections that he wrote on John Searle’s “Chinese room” thought experiment in Chapter 22 of [Hofstadter and Dennett].

Page 308 The term Parfit prefers is “psychological continuity”… See [Nozick] for a lengthy treatment of the closely related concept of “closest continuer”.

Page 309 what Einstein accomplished in creating special relativity… See [Hoffmann].

Page 309 what a whole generation of brilliant physicists, with Einstein at their core… See [Pais 1986], [Pais 1991], and [Pullman].

Page 315 just tendencies and inclinations and habits, including verbal ones… See the Prologue for my first inklings of this viewpoint. See also my Achilles–Tortoise dialogue entitled “A Conversation with Einstein’s Brain”, which is Chapter 26 in [Hofstadter and Dennett], for more evolved ideas on it.

Page 320 Dave Chalmers explores these issues… See [Chalmers]. I always find it ironic that Dave’s highly articulate and subtle ideas on consciousness, so wildly opposed to my own, took shape right under my nose some fifteen or so years ago, in my very own Center for Research on Concepts and Cognition, at Indiana University (although the old oaken table in Room 641 is a bit of a tall tale…). Dave added enormous verve to our research group, and he was a good friend to both Carol and me. Despite our disagreements on qualia, zombies, and consciousness, we remain good friends.

Page 321 with a nine-planet solar system… I’m not about to enter into the raging debate over poor Pluto’s possible planethood (is Disney’s Pluto a dog?), although I think the question is a fascinating one from the point of view of cognitive science, since it opens up deep questions about the nature of categories and analogies in the human mind.

Page 322 Z-people… laugh exactly the same as …Q-people… See “Planet without Laughter” in [Smullyan 1980], a wonderful tale about vacuously laughing zombies.

Page 324 Dan Dennett’s criticism of such philosophers hits the nail on the head… See especially “The Unimagined Preposterousness of Zombies” in [Dennett 1998] and “The Zombic Hunch” in [Dennett 2005] for marvelous Dennettian arguments.

Page 325 you can quote me on that… Actually, the image is Bill Frucht’s, so you can quote Bill on that. I had originally written something about a Flash Gordon–style hood ornament, and Bill, probably correctly seeing this 1950’s image as too passé, perhaps even camp, pulled me single-handedly into the twenty-first century.

Page 326 What is this nutty Capitalized Essence all about? I concocted the phrase “Capitalized Essences” when I wrote the dialogue “Three-Part Invention” in [Hofstadter 1979].

Page 333 for all you know, what I am experiencing as redness… The most penetrating discussion of the inverted-spectrum riddle that I have read is that in [Dennett 1991].

Page 333 Bleu Blanc Rouge… The colors of the French flag are red, white, and blue, but the French always recite them in the order “blue, white, red”. This makes for a tongue-in-cheek suggestion that their color experiences are “just like ours, but flipped”.

Page 339 the so-called problem of “free will”… There had to be some arena in which Dan Dennett and I do not quite see eye to eye, and at this late point in my book we have finally hit it. It is the question of free will. I agree with most of Dan’s arguments in [Dennett 1984], and yet I can’t go along with him that we have free will, of any sort. One day, Dan and I will thrash this out between ourselves.

Page 340 the analogy to our electoral process is such a blatant elephant… This idea of “votes” in the brain is discussed in Chapter 33 of [Hofstadter 1985], as well as in the Careenium dialogue, which is Chapter 25 of the same book.

Page 345 gentle people such as…César Chávez… In the late 1960’s and early 1970’s, deeply depressed by the assassinations of Martin Luther King and Robert Kennedy, I worked intensely for the United Farm Workers Organizing Committee (later known as the “United Farm Workers of America”) for a couple of years, first as a frequent volunteer and then for several months as a boycott organizer (first for grapes, then for lettuce). In this capacity I had the chance to meet with César Chávez a few times, although to my great regret I never truly got to know him as a person.

Page 346 As far as I can peer back… The translation is my own.

Page 347 This was an abhorrent proposal… The translation is my own.

Page 347 a book entitled ‘Le Cerveau et la conscience’… This was [Chauchard].

Page 351 Many performers have been performing… The translation is my own.

Page 361 Riposte: A Soft Poem… There is a method to my madness in this section. In particular, both paragraphs were written to an ancient kind of meter called “paeonic”. What this means is that three syllables go by without a stress, but on the fourth a stress is placed, without its seeming (so I hope) to have been forced: “And yet to you, my faithful reader who has plowed all through this book up to its nearly final page…” One last constraint upon both paragraphs is simply on their length in terms of “feet” (which means stressed syllables). The number of these “paeons” must be forty, and the reason is, I’m mimicking two paragraphs of forty paeons each on page 5a of Le Ton beau.

Page 376 There is a method to my madness… There is a method to my madness in this footnote. In particular, the footnote both describes and represents an ancient meter called “paeonic”. What this means is that three syllables go by without a stress, but on the fourth a stress is placed, without its seeming (so I hope) to have been forced. I now will offer one small sample for your pleasure, and respectfully suggest that you try reading it aloud: “There is a method to my madness in this footnote…” In particular, I’ve got to use exactly forty feet because I’m mimicking two paragraphs of forty paeons each on page three hundred six-and-seventy of I Am a Strange Loop.