Infinite Negative Utility

Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid is a classic text that's had a strong influence on countless people. It's a venerated book among a certain stripe of computer scientist and mathematician, and it's reputed to have inspired generations of people to pursue the study of logic and artificial intelligence. It's a meandering meditation of a number of topics that oscillates around logic, language, formal systems, biology, neurology, music, art, and many more topics, with a particularly strong showing from the three titular figures—the logician Kurt Gödel, the artist Maurits Cornelis Escher, and the composer Johann Sebastian Bach—as well as a heaping dose of writer and mathematician Lewis Carroll, who does not appear in the title but, at least on my copy, is explicitly invoked by the subtitle, “A metaphorical fugue on minds and machines in the spirit of Lewis Carroll.” Many people count it as one of their very favorite books.

So, if you're among the latter group, I should warn you that I'm about to give it a very lukewarm review.

I first read Gödel, Escher, Bach when I was about 20, while I was an undergraduate, and it's important to note that I was double-majoring in computer science and linguistics, and had a particular love of formal systems and logic, but was also proudly a generalist, and had a long-standing love of literature and music. That particular configuration of interests meant that this book was laser-focused to speak to exactly the things that I loved. It was a perfect book for me!

...well, it would have been, but for some reason I kept struggling to get through it. I thought highly of it, but my secret shame was that my admiration for the book was based mostly on the first two hundred or so pages. It took slogging effort to get myself through the rest, effort sporadically applied over the course of years. I did eventually make my way through the whole thing, but even now, I can't necessarily be sure I've read and absorbed every page, even though I've certainly looked at each one. By the end, my impression of the book was much more reserved: it does have some genuine high points and I can understand how it came to have its classic reputation, but I also felt it had a number of problems I've rarely seen discussed. For me personally, it ended up falling short of its reputation as a sparkling, effervescent text that drew together art, mathematics, and culture, and given how little I've seen this discussed, I wanted to write down why I feel this way.

The book is arranged into chapters, each beginning with a dialogue explaining a concept often using imaginative metaphor and a Socratic style, which is followed by a more traditional prose exploration of the concepts introduced in the dialogues. The dialogues are a big part of why I originally struggled with the book: they are meandering and long, and they regularly outstay their welcome. The Socratic style is a difficult one to write well without seeming contrived and difficult, and the book occasionally manages it, but it often falls incredibly flat: usually, they feature one character (usually Tortoise) explaining something verbosely, with conversational asides, but otherwise more or less mechanically, while the other character (usually Achilles) simply responds, “I see! Aha!” and follows up with a question that no real learner would ask but happens to be the next thing Hofstadter wants to talk about.

The other sections revisit the same ideas in prose, giving more concrete examples and dispensing with the catechistic form, and as such are able to give much terser and more interesting examples. Many of these are much clearer, and I remember on my first attempt at reading the book I was often tempted to skip the dialogues and read those first, because their explanations were often much more satisfying and took only a fraction of the time to read through. In some cases, the dialogues attempted to use metaphors that were nonsensical or even broken, and only by reading the chapter afterwards did the dialogue make any sense!

A good example here is the book's explanation of Gödel's incompleteness theorem. The high-level, slightly handwavey description of this theorem is that, for any sufficiently expressive mathematical proof system, there are more true facts within the system than there are proofs for facts, which in turn means that not every fact can be proved: in short, that not every mathematical fact has a corresponding mathematical proof. My short explanation papers over a number of important features of this theorem, such as what I meant by a 'sufficiently expressive mathematical system', and all those features are addressed much more rigorously by the actual proof. It's a fascinating theorem, and to Hofstadter's great credit, Gödel, Escher, Bach helped bring knowledge of this theorem to a much wider population.

Unfortunately, when I first began to read the dialogue which touched on the theorem, I was frankly mystified. Hofstadter decided to explain its working by coming up with a metaphor involving record-players and records that are designed to physically break the record-players they're played on. I'm familiar with how record-players work, but I have never played a record designed to break a record-player! This isn't an intuitive metaphor, because while I have intuition for the operation of records and record-players, I don't have any intuition at all about the universal manufacture of record-player-breaking records. The metaphor raises a number of questions: can the problem of record-player-breaking-records be suitably addressed by redesigning record-players? If no, why not? What if we simply read the information from a record using a visual device with no moving parts? What if we...?

A good metaphor has depth: you can convert a situation into the metaphor, visualize or reason about the implications of the metaphorical situation in isolation of the original situation, and then apply that back and have learned something about the original situation. However, the record-players in this metaphor don't actually work like record-players in the real world, so my own lack of intuition means that reasoning about the original situation via the metaphor is effectively impossible. When I first read the dialogue, I had no idea what was being explained: once I started the following prose chapter, I realized that the “record-players” were formal systems, “records” in were theorems designed to be unprovable within those formal systems, and that the whole thing was an awkward physical metaphor for Gödel's incompleteness theorem. In fact, it was only this realization that made me fully grasp the workings of metaphor in the first place: instead of the metaphor illuminating the theorem, I had to use my knowledge of the actual theorem to grasp what Hofstadter intended for the metaphor!

This is a pretty egregiously bad example, but it was also the point in the book where I realized that I wanted to like the book much more than I actually liked it in practice. I began to read onward and reread past sections with more skepticism, and I realized that the weaknesses which were particularly evident in the dialogue about Gödel's paradox were still partially present in many of the other dialogues. The original inspiration for the dialogue chapters was Lewis Carroll's short allegory What The Tortoise Said To Achilles, which expands on Zeno's paradox of motion to make a point about the foundations of logic. Carroll's dialogue is tight and focused and uses a rather clever metaphor, but the dialogues that punctuate Gödel, Escher, Bach are broad and meandering and the metaphors range from moderately serviceable to (like the one above) actively nonsensical, and the writing style is a mostly-mediocre Carroll pastiche, which means the characters often gratingly pepper their dialogue with interjections like, “Oh, my gracious! Oh, dear me! Oh, but you misunderstand! Go-golly! Oh, but that certainly won't do!” I eventually came to the conclusion that, while the dialogues are one of the more memorable features of the book, they're also an active impediment to conveying much of the book's material in an efficient and clear way.

The non-dialogue chapters, as I've said, are better, although they also range in quality. Many of them are clear, lucid explanations of mathematical concepts intended for a layperson, which often begin by introducing mathematical systems through simple examples, showing what can be done with pure symbol manipulation of those systems, and only afterwards pulling back the curtain to explain what they “mean” in a mathematical sense. The explanations of computational systems have a similar quality, although several of the later chapters feel rather too complicated for their comparatively simple conclusions. On the other hand, the topics that aren't about math or computers (or the shorter bits on workings of DNA) are introduced in a disappointingly cursory way that mostly consists of handwaving and pictures. Those latter sections lack depth and often betray strikingly little familiarity with or respect for the topic in question.

To give an egregious but illustrative example: Hofstadter mentions the experimental composer John Cage on a number of occasions, often bringing up Cage's modernist and aleatoric work as a counterpoint to the meticulously tightly-constructed melodies of Bach. Hofstadter is unsurprisingly negative about Cage's work, and usually characterizes it as avant-garde social commentary masquerading as music, and at one point a dialogue wryly suggests that John Cage might belong in a zoo. John Cage is most famous—or most infamous—for his piece 4'33”, which requires that a performer or group of performers walk onto stage and do not play their instruments for four minutes and thirty-three seconds. (It properly consists of three individual “movements” of non-playing whose lengths have been inconsistently specified across various editions of the score.) Hofstadter brings this piece up in a dialogue¹:

Tortoise: [...John Cage] has composed many celebrated pieces, such as 4'33”, a three-movement piece consisting of silences of different lengths. It's wonderfully expressive—if you like that sort of thing. Achilles: I can see where if I were in a loud and brash café I might gladly pay to hear Cage's 4'33” on a jukebox. It might afford some relief! Tortoise: Right—who wants to hear the racket of clinking dishes and jangling silverware?

Tortoise's (and Hofstadter's) explanation of Cage's piece is fairly typical of explanations given of the piece: that is, four minutes and thirty-three seconds of silence. But this is, according to Cage's intentions, a strictly incorrect interpretation of what he was trying to do! Cage's actual intention in creating 4'33” was not to depict pure silence, but rather to force listeners in an auditorium to pay attention to the quiet and subtle sounds which they usually ignore when they listen to music. To make this painfully explicit, here is a quote from Cage about the original premiere of 4'33”:

They missed the point. There's no such thing as silence. What they thought was silence, because they didn't know how to listen, was full of accidental sounds. You could hear the wind stirring outside during the first movement. During the second, raindrops began pattering the roof, and during the third the people themselves made all kinds of interesting sounds as they talked or walked out.

Consequently, when Achilles and Tortoise agree that they'd rather hear silence than the sounds of a café, they're getting the point of 4'33” exactly backwards: a performance of 4'33” in a café should ideally compel you to listen to the “racket of clinking dishes and jangling silverware” with more awareness than usual!

As I said, this isn't an uncommon misunderstanding of John Cage's intentions, and reasonable people can and do differ as to whether 4'33” is a reasonable execution of that intention, or if that intention is reasonable in the first place. However, even with a charitable reading of Gödel, Escher, Bach, it's clear that Hofstadter isn't disputing Cage's artistic intention: instead, he doesn't seem to know what sort of artistic intention Cage actually has, preferring to read his own ideas about social commentary into Cage's work. His understanding of Cage and of the musical context in which Cage works is marked by a lack of context, a lack of deep engagement with the ideas there, and most importantly, a lack of respect. In the preface to my edition, he claims² that he had,“...unambiguously heaped scorn on Cage's music, albeit in a somewhat respectful manner,” but there's very little respect or willingness to meet Cage on Cage's own terms here, only guarded derision, and that lack of engagement ends up weakening every section that tries to discuss John Cage in particular and modernist music in general.

This sort of cursory engagement with the cultural features of the book ends up undermining one of the book's major selling points: I had originally seen it as the work of a polymath effortlessly weaving fields together into a multifaceted but uniform whole, but in reality, areas that are more than a step or two outside Hofstadter's areas of expertise (computer science, formal logic, some of the more mathematically rigorous bits of cognitive science) are at best shallow, and at worst are “...heaping scorn...” on things Hofstadter doesn't understand and doesn't appear to want to understand.

Despite the relatively surface-level interaction Hofstadter has with the world outside of mathematics and computers, he nevertheless loves to drop in thick, multilayered references to such topics in every cranny he can find. The central two characters in the dialogues are Achilles and Tortoise, borrowed directly from Lewis Carroll's story above (which in turn borrowed them from the famous paradoxes of the Greek philosopher Zeno), and a Crab and a Genie show up on occasion as well. Their names are regularly abbreviated to a single letter, which means you can't help but notice that those letters happen to map to the names of the nucleobases that appear in DNA—adenine, thymine, cytosine, and guanine. All the dialogues have sing-song names that are usually inspired by music, such as Sonata for Unaccompanied Achilles or Canon by Intervallic Augmentation or Birthday Cantatatata. Off-hand mentions of people and places are often wry and unexplained cultural allusions: a typical example is that, at one point in a conversation about popcorn, Tortoise awkwardly shoehorns in the story of a “Schönberg factory” in Vienna that outraged consumers by stopping production of a delicious tonic in favor of a boring cereal, this being a nod to the Viennese composer Arnold Schoenberg's transition from traditional melodic compositions to experimental atonal pieces, a nod that never comes up elsewhere in the text of merits any explanation.³

I will admit right away that I personally find these heaps of unnecessary nods more tedious than interesting or endearing. There's a lot of reference, but very little of it means anything. Occasionally, a dialogue will use these references to actually explain something—one dialogue is written in the form of a crab canon, and thus is identical when read forward or backward, as a memorable way of explaining that form of musical canon—but most of the musical or biological or literary allusions are there really for their own sake. These things are rarely being commented on, or discussed in interesting context, or connected to other ideas. Instead, these ideas simply appear because this is a book in which ideas appear, interrupting the text for a shoehorned cameo, like Stan Lee in a comic book movie. Why do the characters' names map to nucleobases? I suspect if you asked Hofstadter, he'd claim it's because one of the themes of the book is that all these ideas are connected (in the titular “golden braid”), but this kind of reference doesn't actually connect anything to anything: it merely presents things adjacent to each other. It's all flavor, no substance.

This might seem unfair, so I'll give a very specific but pervasive instance of this sort of meaningless flavor. Many parts of the book invoke Zen, which is a school of Buddhism that originated in China and has spread to several other Asian countries but, in the Western mind, is usually associated with Japan. (We do, after all, know this school by its Japanese name Zen and not by its Chinese name Chán, its Korean name Seon, or its Vietnamese name Thiền.) Hofstadter's idea of Zen is a substance-less cliché: it consists almost entirely of faux-Eastern “Oriental” aesthetics and some handwaving about kōans (which are stories used in Zen practice for teaching and meditation) without really delving into any particular aspect of actual Zen thought or practice. There are lots of references to it—for example, he names a theorem MUMON, after the Japanese name of Zen master Wúmén Huìkāi, as part of a vague and largely pun-based connection to one of Wúmén's collected kōans—but none of those references have any substantial connection to the history or practice of Zen. In reality, Zen is a religious sect with history and cultural context and a complicated, multifaceted conversation that has been carried on throughout centuries. In Hofstadter's telling, Zen is just some funny nonsensical stories from Japan.

The edition I have includes a preface in which Hofstadter talks, twenty years after the book's release, about the book itself, its reception, and its legacy, and he goes out of his way to complain about a negative review of the book which accused him of being a hippie trying to popularize Zen. Hofstadter objects to this review because

As I declare at the start of Chapter 9, I find Zen not only confusing and silly, but on a very deep level utterly inimical to my core beliefs. However, I also find Zen's silliness—especially when it gets really silly—quite amusing, even refreshing, and it was simply fun for me to sprinkle a bit of Eastern spice into my basically very Western casserole. However, my having sprinkled little traces of Zen here and there does not mean that I am a Zen monk in sheep's clothing.

In this passage, Hofstadter openly admits to exactly the charge I'm bringing: that his inclusion of Zen is all about clichéd aesthetics (“Eastern spice”) and not at all about any of its substance—in this case, because he apparently doesn't seem to think it has any!

What Hofstadter doesn't admit to, but what I would argue, is that the whole book does this with almost every non-mathematical topic it tackles. His explanations of mathematics-adjacent topics do have substance and are often reasonably well-explained, but every time he branches out, he doesn't seem to realize that he's regurgitating shallow, half-misunderstood cliché: his discussions of modern art and music are, as I mentioned before, deeply lacking in this regard, but he name-checks plenty of artists, musicians, and writers with a high school understanding of who they were and what they did, preferring to pepper the text with photos of wacky paintings, drawing he made of letters that are made up of other letters, and tales of half-understood kōans. They're all spice: his “casserole” is a few insubstantial layers of food underneath inch-thick layers of spices.

This also presents a problem with the entire underlying program of the book: it's supposed to present examples of a common important idea—self-reference—resurfacing throughout various disparate areas, including mathematics and computation and art and music, but while this idea is well-motivated in the parts about mathematics and computation, but because most of the other topics the book tackles end up being just shallow aesthetics, then the “deep connections” there can only be present in shallow aesthetic ways. This was, for me, the ultimate breakdown of the promise of the book, as the grand unifying theme—the titular “eternal golden braid” of self-referential structures across domains—was only capable of unifying a few problem domains, as the rest of those connections were pretty but ultimately insubstantial.

While rereading bits of the book in order to write this post, I flipped through it at random and came across the photos marked as Figure 81, which in my copy at least appears on pages 490 and 491. These pages contain photos of television screens that are in turn displaying geometric images that result from pointing a camera at the screen: infinite turning shapes, nested and sweeping frames, eventually deforming into twirling light patterns. They are described in captions, beginning with plain descriptions like, “What happens when you rotate the camera,” and gradually becoming more florid, with captions like, “The galaxy has burned itself out, and become—a black hole!” The actual caption beneath these photos says the following:

Twelve self-engulfing TV screens. I would have included one more, had 13 not been prime.

These photos are fun! They feel especially endearing in 2018 because of the late-70's television depicted, and they depict a fun experiment that I did as a child as soon as I got my hands on my parents' bulky camcorder⁴. The captions, however, add very little, and the final comment (“...had 13 not been prime”) includes a bit of extra unnecessary whimsy that seems to wink at the reader but adds absolutely no meaning. Like so much of the book, it seems to hint at something grander while not signifying anything in particular. The photos themselves might be a fun illustration of something, but they're not a particularly deep illustration of anything, and their inclusion here (surrounded by several pages of Achilles and Tortoise pontificating about the notion of “self-engulfing”) doesn't bring any more enlightenment than when I first pointed a camcorder at a TV when I was four.

“A fun illustration of something,” is pretty much as far as the book goes: it hints at grand unifying patterns, but the pattern it finds is just the abstract notion of self-reference, and then it keeps bringing it up, making a few unnecessary references, showing some pictures, and asking, “Isn't that cool? Isn't that weird?” It'll give a perfectly competent (if somewhat verbose) description of formal systems, but as soon as it tries to venture connections to other domains, or to explain more complicated or nuanced details, it turns out that the only connections it can draw consist of vigorous handwaving. The whole book boils down to Hofstadter giving a competent lecture on logic, intimating the existence of an eternal golden braid, and then pointing at some fun photos of televisions.