A discarded review of ‘Godel, Escher Bach: an Eternal Golden Braid’
Recently I began to write a review of Hofstadter’s Godel, Escher, Bach, until I realized that the book defied summary more than all the other books I had previously said “defied summary.” Thus, I gave up on reviewing the book after not too long. I present my discarded review below just in case it motivates someone else to pick up this masterful tome and let it enrich their life.
Of Hofstadter’s GEB, Eliezer once wrote:
This is simply the best and most beautiful book ever written by the human species...
I’m not alone in this opinion, by the way. For one thing, Gödel, Escher, Bach won a Pulitzer Prize. Or just pick a random scientist and ask ver what vis favorite book is, and 1 out of 5 will say: “Gödel, Escher, Bach”. No other book even comes close.
It is saddening to contemplate that every day, 150,000 humans die without reading what is indisputably one of the greatest achievements of our species. Don’t let it happen to you.
Sure, if you’re just an average person, you might not understand everything in this book—but when you’re done reading, you won’t be an average person any more.
It’s easy to see GEB’s effect on Eliezer’s writing: the “concrete, then abstract” pattern, the koans, the puzzles, the conversational coverage of technical concepts in math and computer science… it’s all here in spades in GEB.
What GEB Is
In the preface to the 20th anniversary edition, Hofstadter clarifies what GEB is and is not. It is not about how reality is “a system of interconnected braids.” It is not about how “math, art, and music are really all the same thing at their core.” Instead, says Hofstadter:
GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter… GEB approaches [this question] by slowly building up an analogy that likens inanimate molecules to meaningless symbols, and further likens selves… to certain special swirly, twisty, vortex-like, and meaningful patterns that arise only in particular types of systems of meaningless symbols. It is these strange, twisty patterns that the book spends so much time on, because they are little known, little appreciated, counterintuitive, and quite filled with mystery [that] I call… “strange loops”...
...the Godelian strange loop that arises in formal systems in mathematics… is a loop that allows such systems to “perceive itself,” to talk about itself, to become “self-aware,” and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self.
...the shift of focus from material components [of the human mind] to abstract patterns allows the [surprising] leap from inanimate to animate, from nonsemantic to semantic, from meaningless to meaningful, to take place. But how does that happen? After all, not all jumps from matter to pattern give rise to consciousness or soul or self… What kind of pattern is it, then, that is the telltale mark of a self? GEB’s answer is: a strange loop.
The irony is that the first strange loop ever found… was found in a system tailor-made to keep loopiness out… Bertrand Russell and Alfred North Whitehead’s famous treatise Principia Mathematica...
...For the French, the enemy was Germany; for Russell, it was self-reference. Russell believed that for a mathematical system to be able to talk about itself in any way whatsoever was the kiss of death, for self-reference would… necessarily open the door to self-contradiction...
Kurt Godel realized that… self-reference not only had lurked from Day One in Principia Mathematica, but in fact plagued poor PM in a totally unremovable manner. Moreover, as Godel made brutally clear, this thorough riddling of the system by self-reference was not due to some weakness in PM, but quite to the contrary, it was due to its strength. Any similar system would have exactly the same “defect.”
[Godel had discovered that] any formal system designed to spew forth truths about “mere” numbers would also wind up spewing forth truths… about its own properties, and would thereby become “self-aware,” in a manner of speaking.
[But] strange loops are an abstract structure that crop up in various media and in varying degrees of richness.
A Musico-Logical Offering
Hofstadter opens with the story of J.S. Bach’s Musical Offering for King Frederick, which contains a particular canon that sneakily shifts from one key to another before its apparent conclusion, and when this modulation is repeated 6 times, the piece ends up at the original key but one octave higher. This is our first example of a “Strange Loop”:
The “Strange Loop” phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some heirarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.)
Other examples occur in the drawings of M.C. Escher, for example this famous one.
The liar’s paradox (e.g. “This statement is false”) is a one-step Strange Loop. Related to this is a Strange Loop found in the proof for Godel’s Incompleteness Theorem, which states, roughly:
All consistent axiomatic formulations of number theory include undecidable propositions.
Before Godel, Russell and Whitehead tried to banish Strange Loops from set theory in Principia Mathematica. But Godel’s theorem showed
...not only that there were irreparable “holes” in the axiomatic system proposed by Russell and Whitehead, but more generally, that no axiomatic system whatsoever could produce all number-theoretical truths, unless it were an inconsistent system!
The goal of the book is to explain these Strange Loops in more detail, and how they may explain how animate beings arise from inanimate matter.
Meaning and Form in Mathematics
After a tutorial on formal systems, Hofstadter argues that
...symbols of a formal system, though initially without meaning, cannot avoid taking on “meaning” of sorts, at least if an isomorphism is found.
The vast majority of interpretations for a formal system are meaningless, but if an isomorphism can be found between the formal system and some piece of reality, that isomorphism provides the symbols their “meaning.”
But you may discover multiple isomorphisms, and thus the symbols of a formal system may have multiple meanings. It makes no sense to ask, “But which one is the meaning of the string?”:
An interpretation will be meaningful to the extent that it accurately reflects some isomorphism to the real world.
...
GEB is great as many things; as an introduction to formal systems, self reference, several computer science topics, Gödel’s first Incompleteness Theorem, and other stuff. Often it is also a unique and very entertaining hybrid of art and nonfiction. Without denying any of those merits, the book’s weakest point is actually the core message, quoted in OP as
What Hofstadter does is is the following: he identifies self-awareness and self-reference as core features of consciousness and/or intelligence, and he embarks on a long journey across various fields in search of phenomena that also has something to do with self-reference. This is some kind of weird essentialism; Hofstadter tries to reduce extremely high-level features of complex minds to (superficially) similar features that arise in enormously simpler formal and physical systems. Hofstadter doesn’t believe in ontologically fundamental mental entities, so he’s far from classic essentialism, yet he believes in very low level “essences” of consciousness that percolate up to high-level minds. This abrupt jumping across levels of organizations reminds me a bit of those people who try to derive practical everyday epistemic implications from the First Incompleteness Theorem (or get dispirited because of some implied “inherent unknowability” of the world).
Now, to be fair, GEB considers medium levels of organization in its two chapters on AI, but GEB’s far less elaborate on those matters than on formal systems, for instance. The AI chapters are also the most outdated now, and even there Hofstadter’s not really trying to do any noteworthy reduction of minds but instead briefly ponders then-contemporary AI topics such as Turing tests, computer chess, SHRDLU, Bongard problems, symbol grounding, etc.
To be even fairer, valid reduction of high-level features of human minds is extremely difficult. Ev-psych and Cognitive Science can do it occasionally, but they don’t yet attempt reduction of general intelligence and consciousness itself. It is probably understandable that Hofstadter couldn’t see that far ahead into the future of cogsci, evpsych and AI. Eliezer Yudkowsky’s Level of Organization in General Intelligence is the only reductionist work I know that tries to wrestle all of it at once, and while it is of course not definitive or even fleshed-out, I think it represents the kind of mode of thinking that could possibly yield genuine insights about the mysteries of consciousness. In contrast, GEB never really enters that mode.
Agreed on the weakness of Hofstadter’s core message, but for another reason. While self-awareness is one feature that is often referred to as “consciousness”, it isn’t the biggest trouble-maker. The “qualia” of experience can occur without self-awareness, and probably even in organisms utterly lacking in self-awareness. The subjective feels have been the biggest stumbling block to philosophical progress, and turning away from them (or worse, thinking that they amount to a kind of self-reference) doesn’t help the reductionist cause.
Is there a citation on this statistic? Almost none of the scientists I know have even heard of GEB, let alone read it; ~10% know of it, and less than half of them have read it. (Granted I hang out with a lot of atomic physicists, so my sample may be biased).
It would have been more accurate to limit the sample to mathematicians and computer scientists, which I think was lukeprog’s (subconscious) reference class.
That I could see...the figures I can find say 400,000 copies sold. Assuming half of those are to mathematicians and computer scientists, that’s 200,000 sales to our reference class, which would be reasonable once we take into account people borrowing/downloading the book.
Wow, I had no idea Eliezer loved it that much. I own it, and I once read through a few chapters, but it was mostly a collection of (albeit very intelligent) puzzles and ideas that I was familiar with, so I stopped. Am I missing something, or am I just already not “an average person”?
I read the whole book and also felt meh about it. It might feel different when it’s your first introduction to the ideas, though. I remember being very impressed by Strategy of Conflict because the ideas in it were new to me.
I read the book in high school and loved it. It was my first introduction to the ideas. And there was an unusual extra psychological factor boosting my interest. One of my math teachers, a brilliant, very eccentric guy once saw the book in my hand, and started to shout at me very-very loudly in front of a crowd about how evil this book was. It was a crazy scene. He had a serious problem with reductionism.
This teacher taught us ultrafilters and Löwenheim-Skolem when we were 17, but he also told us that set theory is false. I confronted him: if set theory is false, surely one of the axioms must be false, so which ones does he object to? He told me the whole thing is stupid. This didn’t satisfy me, so I kept asking, and finally he said that for example, the pair axiom is false. It tells us that we can put things into pairs without this affecting them in any way. If I was put together into a pair set with a beautiful woman, and I wasn’t affected by this, that would mean that I am impotent. Set theory makes mathematics impotent. I didn’t completely buy his story on set theory, but it definitely influenced my thinking somewhat. On the other hand, I chose to ignore his outburst against reductionism.
Without revealing my grounds (except that I’ve known many mathematicians), I would bet at even odds that your high-school math teacher grew up behind the Iron Curtain. Am I right?
You are very right. I am from Hungary. The Iron curtain fell exactly the year when my GEB story took place. The guy was a promising young mathematician before becoming a high-school teacher of gifted students at the famous Fazekas high school. Although he was never bitter about it at all, I suspect this change of course was somehow related to the fact that he was a sympathizer of the underground democratic opposition.
Excellent- I’d actually assumed you had grown up in the English-speaking world and that you just happened to have an Eastern European teacher for some reason, even though that’s a much less likely way for it to happen. Still, it’s nice to see I can trust my instincts about the national character of particular mathematical eccentricities- something about the style of the example reminded me strongly of Erdős (except for the personal irony it would have had for him).
I felt that, when I read the book the first time and hadn’t encountered many of the concepts before, his way of introducing the ideas was hard to understand. After I had been introduced to the same ideas in a more traditional way, I didn’t think his explanations told me anything that I didn’t know.
(For the most part, that is—I do seem to recall that his explanation of inductive proofs allowed me to get the concept a little better, after I had already had the concept explained to me elsewhere.)
I’ve tried reading it a couple of times, managing to read maybe about one third or one half. When I first read it, many years ago, it was good at really teaching the notion of formal systems and the fact that they were really only governed by their own axioms and rules, but that’s about all that I got out of it.
Isn’t that true for any book that some people like it and others not so much, and that it depends on what the person in question is already familiar with? I liked GEB because it was an easy to read good explanation of basic ideas of formal systems and some other stuff as well. If I were already familiar with the ideas, I wouldn’t probably finish reading it. But that doesn’t mean the book is bad, in the same way as the fact that I don’t laugh at a joke I have heard ten times before doesn’t mean the joke is bad.
I didn’t see what the fuss was about either. Keep meaning to go back and give it another chance...
The second half of the book is of a very different nature than the first, though it’s still quite possible you’d be familiar with it. If you have the generalized anti-zombie principle firmly in hand in advance, it will be a bit less world-changing.
Forgive my ignorance, not having read GEB, but I can’t help being underwhelmed by the Bach example. This Youtube video plays the Bach canon in question. The canon begins in C minor and modulates up in whole tones until it arrives at C minor again an octave higher (the Youtube recording returns to the same octave, but it does so using trickery—notice that in Bb minor, the sixth time through, it ends on a D notated a ninth above the next C but sounding only a step above it).
Unless I’m missing something, this is rather like saying that if you walk for ten hundred-metre lengths, you’ll end up a kilometre from where you started. Yes, you will, but so what?
I believe this is one of the weakest possible examples from the book… though it may be one of the few that can actually be extracted from context without requiring dragging in a great deal more. I would have gone with one of the Escher examples, as it’s quicker to grok than Gödel, and the Bach examples are, well, a bit of a stretch.
Complete traversals of a set of whole-tone-related keys are incredibly rare in music this early; I don’t know of another example and would not be that surprised if one doesn’t exist. You’re right that such a piece seems easy in principle, but that’s from a current viewpoint. So I think that Hofstadter may in part be implicitly relying on some knowledge of the early-eighteenth-century context when he stresses how unusual the piece is.
(There is actually also a technical reason why this loop is so strange—it has to do with what music theorists call “crossing the enharmonic seam”—but Hofstadter doesn’t appear to be referring to that.)
It is a neat trick, and not something that happens often, but I would guess that’s because it’s not useful as anything other than a neat trick. I’m not seeing the eternal golden braid in it, is all.
Actually, if Bach had kept the pattern intact without “crossing the enharmonic seam” it wouldn’t be much of a loop at all; the piece would end up in B# minor after six repetitions.
(edit: sp.)
Yeah, I’m in agreement with you and others that it isn’t the most compelling example he could have chosen.
As to the enharmonic seam thing, that is indeed the point: you either have to cross the enharmonic seam by spelling two identical-sounding intervals differently (in this case, one of the major seconds has to be spelled as a diminished third) or else you have to deny the seeming aural fact of octave equivalence by spelling the return of C as B-sharp. Since composers are extremely reluctant to do the latter, they have no choice but to do the former—a commonplace in the nineteenth century, a bit of a special trick in the mid-eighteenth.
If I recall correctly, he focuses on the fact that the piece may be played in a cyclical fashion, allowing an infinite loop of sorts.
What’s special or interesting about a musical piece that can be played cyclically? Such a piece is easy to compose by editing the two ends to align, even without reference to whatever is in the middle of the piece.
In the Bach example, if you go up an octave on every loop, you can’t play forever anyway (within human hearing).
It’s not that hard to do (somewhat harder if you want to keep your Bach-style harmonies intact), and I don’t think anyone claimed it was that hard, simply that it induces an interesting self-referential cycle. There’s something rather amusing (at least to me) about a piece of music that can be played an infinite number of times without repeating a musical phrase more than the few times it occurs in a single cycle.
As for the quick rise out of the human range of hearing, it’s just a small side effect that prevents musicians from getting caught in an infinite loop.
So GEB’s entire point here is that some infinite sequences of similar-but-different objects have self-referential formulations?
Just like these are equivalent: a(n+1)=a(n)+2; a(0)=0 vs: a(n)=2n for all n
Each element has the structure of “an even integer”, but the first form is self referential while the second one isn’t.
I fail to see a deep meaning in this, or any similarity with consciousness. Can someone enlighten me? Did I merely take the book’s example out of context?
Nope, you’ve got the right idea about his example. It occurs early on in the book, while he’s trying to explain simple concepts to readers through non-technical analogy; sort of the way he explained complement spaces to readers by first asking which word contains the sequence “ADAC” in order (headache), and then asked what word contains the sequence “HEHE” in order; nothing particularly special about that either, but it teaches the reader a useful trick without presenting it mathematically.
GEB has been sitting on my shelf for almost a year. I didn’t realize I might be missing out that much. Starting it today.
My most valuable possession is a signed copy of GEB given as a graduation present. Hofstadter wrote “Congratulations on your graduation from Harvard—aka the Harvard of the West of the East”. Hahahaha … okay, sorry.
If you read some of his later work with the Fluid Analogies Research Group, you may find that even funnier.
The Feynman Lectures?