On (Finally) Finishing Gödel, Escher, Bach

September 28, 2018

(From February 2017)

Yesterday I finally finished Douglas Hofstadter’s Gödel, Escher, Bach. It’s been a long time coming—I started the book (and made it maybe halfway through) back in 2013, then started again in 2015 and made it no farther. For whatever reason, this time around I gathered enough momentum to finish the thing.

GEB reads like a grad-school seminar on most of twentieth-century intellectual history, taught by a professor who’s brilliant but unapologetically wacky. It alternates between dense, serious chapters and whimsical dialogues between a cast of Lewis Carroll–inspired characters: Achilles, the Tortoise, and new friends like the Crab, the Anteater, and Charles Babbage.

The book’s basic premise is that:

  • Formal systems (formal logic, typographical number theory, AI) acquire meaning despite being made up of meaningless low-level components.
  • Sufficiently powerful formal systems fail to capture the entire reality they’re intended to mirror—there will always be real or true elements that fall outside the system.
  • In the case of AI and brains, the “self” arises from a “strange loop,” or instance of indirect self-reference, between one level of the formal system and another.

That’s the highest-level summary. One step lower, GEB plays with what feels like dozens of themes from logic, mathematics, music, art history, computer science, molecular biology, and even Zen Buddhism. The above list might be expanded into this one:

  • Things can be represented in multiple ways, at multiple levels of abstraction. For example, a piece of music can be expressed as physical wax (a record) or as sheet music. The wax is a fairly low-level (meaningless) representation and the sheet music is at a higher level (more meaningful); the same music could perhaps be expressed yet one level higher in terms of themes, emotions, or schools of composition.
  • Different levels, and indeed completely different things, preserve information between themselves in what are called “isomorphisms.” Isomorphic things represent corresponding information when viewed at a sufficiently high level, but are completely different at lower levels: the record and the sheet music are unlike each other at the level of wax and paper, but they express the same melody and are therefore isomorphic. And isomorphisms themselves can take place at varying levels: to borrow an example of Hofstadter’s, a book by Dostoyevsky (in the original Russian) might be considered isomorphic to a crude word-for-word English transliteration, to a literary translation whose translator has taken significant artistic license, or to a work by Dickens—at an extremely high level, the “same sort of book” in English.
  • Isomorphisms are what allow formal systems to (mostly) mirror the world. Formal logic is isomorphic to claims about truth. Typographical number theory is isomorphic to mathematics. But formal systems are incapable of capturing the whole of their domains, because sufficiently powerful systems eventually become capable of indirect self-reference, which leads to paradox. At the simplest level, this looks like the Cretan Epimenides’s claim that “All Cretans are liars.” At the highest level, it looks like Gödel poking holes in Bertrand Russell’s Principia Mathematica by demonstrating that any string in typographical number theory can be translated, via an arbitrary mapping (“Gödel numbering”), into a form that makes indirect, unprovable assertions about typographical number theory itself.
  • The human brain is a system with (it would seem) a vast number of levels between the lowest—mere neural switches—and the highest: emotions, beliefs, memories, self. Hofstadter argues, though the argument here is more hand-wavy, that the sense of self must arise from some level-crossing self-reference within the brain, and further that any successful AI (a sort of formal system) will rely on a similar strange loop.

I found most of the book quite dense (and sometimes detailed to a fault), especially the content dealing with formal logic. (That’s more a statement about my personal education than it is about the book itself, of course.) On the other hand, the dialogues, which are thematically linked to their partner chapters, are marvelous—accessible, playful, and evocative. You could think of the bulk of the book as the price of admission for really appreciating those dialogues, though I’m certain I still missed quite a bit.

Looking back on it, the real value of GEB for me isn’t the content itself—there’s so much of it (and, for me at least, it gets so dense) that the specifics can be hard to remember. (Hofstadter must have spent 40 pages on the exact mechanism by which DNA transmits information, but I couldn’t explain it to you now.) Instead, the value is in gradually picking up Hofstadter’s habits of mind. He hops easily from discipline to discipline, identifying unexpected parallels and subtle paradoxes, all with an infectious sense of joy. You come away from the book with a new vocabulary, an enhanced interest in the seemingly dry disciplines that Hofstadter unites, and a keener sense for self-reference, isomorphism, and paradox.

P.S. After finishing the book, I thought about what a marvelous thing it is to read something so rich. Then I thought about what a waste it is to read books your whole life, fill your brain with these treasures, and die and lose it all. Then I thought about how Hofstadter couldn’t have written GEB without reading dozens of other books, and how—though a single book can’t preserve the low-level information of many—it can nevertheless pass along their high-level essence, preserving information even once the reader/author’s brain is dead. From there follows a notion of reading and writing as a sort of recursive function where each generation reads those that preceded it, then writes books at a higher level of abstraction, such that the quality of books (and of knowledge) rises over time. Perhaps the reader/author’s role is nothing more than serving as a sort of petri dish for growing new books, and the books you write are deterministically calculable from the books you’ve read. (If this is the case, of course, it is every reader’s duty to eventually write.)