I Am a Strange Loop
Page 50
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 a
uthors 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.