Title: Gödel, Escher, Bach – an Eternal Golden Braid
Author: Douglas Hofstadter
Published: 1979, I read the 1999 20th Anniversary edition
This was a daunting book to begin, and no amount of reading reviews beforehand really gave me a feel of what this book was actually like or what it’s about.
I read the 20th anniversary edition, which includes an introduction from the author where he reflects on the creation of the book, the responses to it and more directly addresses what it is actually about, a point which seems to have been a source of much confusion and contention among reviewers. The author also provides some clear introduction on what it is definitely not about (eg the notion that maths, music and art are all one and the same). Having actually read it now, it makes much more sense that many reviews or the introduction would suggest.
Firstly, it’s not as heavy going as it first seems. The main chapters themselves can get into considerable depth, but a conversational tone is maintained throughout and it’s broken up with illustrations (particularly Escher of course, but others as well including many drawings by the author), exercises to help demonstrate the points, and the dialogues between Achilles and the Tortoise, as well as a host of other characters, ranging from a Crab to Charles Babbage and Alan Turing.
Ultimately, the book is mostly not about Escher or Bach, their works are used as a metaphor to illustrate many of the points. Goedel features more centrally but it’s not really about him either.
So what is it about?
Between 1910-12 Alfred North Whitehead and Bertrand Russell published a series of papers titled Principia Mathematica (not to be confused with Newton’s work of that name), attempting to create a formal axiomatic system capable of representing all mathematics through symbolic logic. In 1931, Goedel blew that apart, proving that not only were there elements of PM which were formally undecidable, but that any formal system that was sufficiently powerful to represent all arithmetic would suffer the same defect, this is known as Goedel’s incompleteness theorem.
The author illustrates this theorem in remarkable detail, and uses it a starting point for a rich and metaphorical meditation on the mind and systems of thinking, showing how meaning operates on many levels and higher levels of thinking and meaning can then impact the lower levels, whether we are talking about lines that become letters than become words that become novels, or the firing of neurons in the mind to the higher levels of consciousness, or the firing of gates in a computer sending 1s and 0s to basic code languages, to complex programs and the possibility of artificial intelligence.
As the author himself outlines in his 20th anniversary introduction, the main thesis of the book is how ‘animate beings can come out of inanimate matter’ and how meaningless symbols acquire meaning despite themselves.
All of this is illustrated with a rich array of metaphors, dialogues, allusions, paradoxes and plenty of puns. The mind-bending art of MC Escher, and the intricately composed works of Bach, then serve as metaphors, but incredibly powerful and apt ones that are punned and riffed on in the dialogues and then explained more logically in the subsequent chapter. Other artists, notably Magritte, are also featured, as well as the post-modernist composer John Cage as a counterpoint to Bach, and Zen and its baffling and paradoxical koans.
Each chapter alternates between an introductory dialogue between Achilles and the Turtle (borrowed from Zeno, via Lewis Carroll), where some of the key points of the upcoming chapter are explained in simpler allegories mixed with wordplay. For example Goedel’s incompleteness theorem is explained by the concept of a record player that claims it can play any record (Principia Mathematica) Goedel showed that for any record player of sufficient fidelity there could exist a record with just the right sounds which could vibrate apart said player, so there is no record player that can play any record. (If the record player is not sufficiently high fidelity it may not reproduce the sounds accurately enough, so then cannot meet the claim to be able to play all records either).
The author creates a number of simpler systems to illustrate how a formal system works, and where is doesn’t, showing that it is necessary to step outside of the system to fully understand it, and there are truths that are axioms within the system, but there are also truths about a system which are outside of that system. He shows how such an argument has been put forward to by some to show how true artificial intelligence is not possible, but then goes on to show how these statements are no less true if applied to the human brain.
Recursion plays a key role, particularly as he shows how attempts to create a more robust and Goedel-proof system simply push the problem up another level but still have the same weakness. Through metaphors such as an infinite series of Djins who refer a problem up to the next level of GOD, which stands for God Above Djin, to demonstrations of computer programs that could go on calculating forever without being able to determine something is impossible, ie Turing’s halting problem.
The author likens Goedel’s incompleteness theorem to the Epimenides paradox, where the Cretan says ‘all Cretans are liars’ or in its more basic form ‘This statement is false’. Which led me to wonder from time to time, as Achilles says to the Tortoise at one point, “I can’t quite tell if we’re arguing about something utterly petty or something deep and profound” I did ponder that were I a commissioning editor reading this as a manuscript in the late 1970s, I’m not certain I would have seen it as the idiosyncratic masterpiece that it is, without a Pullitzer prize and 40 years of good reviews behind it. There’s a few hypotheticals involved in that scenario, but it reminded me of a time about 15 years ago when I did get a manuscript from a friend and colleague where he was genuinely convinced he had come up with the Grand Unified Theory of Everything. I had to call in some outside advice to double check but alas he had not succeeded in doing so.
Another fascinating parallel is drawn not with Escher or Bach but with Crick, and the author turns the process of DNA replication into another numerical system and shows how it functions as a self-replicating system which creates new meaning when combined with RNA and proteins.
What i learned
It’s not just a book about Artificial Intelligence, but as a philosophical reflection on the nature of minds and intelligence it has been very influential over the past 40 years in defining what AI is and isn’t. the author notes the progress AI has made in games of checkers, but reflects that mastery at Chess or Go is a long way off (as it was at the time of publication). Of course so many of the tests and levels of AI postulated have since been solved, from speech or text recognition, real-time translation, self-driving vehicle, but at the same time we are not yet close to achieving general AI. As the author himself writes, each step towards AI in some ways shows what intelligence is not.
He also powerful rebuts the notion that because computers are programmed, and can only do what they are programmed to do, that they cannot be intelligent. For example, if a person is making a decision between two things, their individual neurons aren’t also hesitating in this decision, they are firing as they should to create the higher level process of making a decision, and so it could be with a computer as well.
Both brains and computers operate on multiple levels, with each level unaware of the levels below, but dependent on them and the higher levels directly impacting lower levels while still being driven by them. Just as brains can think about themselves, computer programs can write parts of themselves. Neither can fully step outside itself, without stepping outside the system, like Godel’s incompleteness theorem, or Escher’s image of two hands drawing each other. This creates strange loops – which we are all, and is the topic of a subsequent work by Hosftadter, which I look forward to reading soon.
Should you read it?
Ultimately, I hope I’ve given a clearer idea of what this book is like that many of the other reviews, but at the same time I’ve demonstrated how hard it is to articulate these complex topics in such a short space. Goedel Escher Bach is one of the great non fiction books where not a word is wasted and it could not have been any shorter without losing much of its meaning, so for a far better exploration of the above than I have managed, it must be read.