I Am a Strange Loop (74 page)
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
n
th 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.