Notes on GEB


Notes on Douglas Hofstadter's 1979 book Gödel, Escher, Bach: an Eternal Golden Braid — or GEB, as everyone calls it.

Written in Hofstadter's engaging style and playful form, GEB is a unique exposition of Kurt Gödel's remarkable incompleteness theorem, a deep result in mathematical logic, as well as an inspired investigation of the analogies Hofstadter draws from the strange self-referential pattern at the heart of the theorem.

What follows is notes on:

  • What GEB is about,
  • What GEB is not about,
  • How to read GEB,

and a few other things to keep in mind while reading.

On the Preface to the 20th ann. ed.

Everyone interested in GEB should grab the 20th anniversary edition. (Mine is from Penguin Books, ISBN 9-780140-289206)

The book can be quite an overwhelming read, and also notoriously hard to summarise — even for those who manage to make their way through it. Fortunately, Hofstadter did an exceptionally good job at explaining and introducing GEB in the Preface to the anniversary edition.

The Preface is a great starting point for three reasons:

  1. After 20 years of listening to people misunderstand his book, Hofstadter relished the opportunity to write a dense, unadorned explanation of what the book is about and why he wrote it. The Preface is a must read for anyone who couldn't quite put it all together.

  2. It tells the origin story of the book and highlights the extraordinary effort H went to in order to get his thoughts out to the world. It is an illustration of how deeply he feels (or felt) about the subject matter, how personal the work really is. This explains much of the books content and the more colourful stuff.

  3. The Preface also gives the book's working title, which is the key to understanding the book's structure and message. The "mundane" working title was: Gödel's Theorem and the Human Brain.

I'll expand on these items in this piece.

What the book is not about

Before diving into what GEB is about, briefly on the opposite.

Why is GEB such a challenge to accurately summarise? The main reason is that the book touches on so many topics that one can easily lose track of the core ideas. Hofstadter generously expands on each intriguing concept he introduces, and delves quite deep, all the while playing his literary games. There's simply a lot going on, and the unprepared reader can easily get disoriented. And then in our awe of the things we encounter, we end up embracing all kinds of wild explanations from mystical on one extreme to oversimplified and blandly reductive on the other.

Hofstadter gives a few examples of the kind of muted thinking and hyperbolic explanations that fail to capture the book's message:

The first one uses big words, but says very little. The second one is a bit too literal — GEB is no biography. The brief section on Turing at the beginning of Chapter XVIII, a well-placed prelude, serves as an example of the kind of appreciation Hofstadter holds for his intellectual heroes. He would have written a very different kind of book if biographing these names had been his aim.

The fourth example, GEB as a vehicle for mysticism, is quite understandable, but also unfortunate. It is true that with GEB, Hofstadter boldly steps into unknown territory with his theories on the nature of human mind. The reader is then of course equally free to extrapolate as well. In my reading, however, Hofstadter is quite careful to signal when he is talking about things that really are and when he's alluding to things that could be. The real parts are backed by the latest scientific thinking of the day, while the more fantastical and exciting ideas are approriately qualified.

The whole book is about an analogy: drawing parallels between a known thing and something else, something unrealised. That's all.

The third example is the most interesting one, because it has a grain of truth in it. Strange patterns of the kind that get Hofstadter going certainly manifest in each of math(s!), art, and music, and there really are rich interlinks between the three. But to claim that they all share some fundamental essence is a step too far. Perhaps there is a secret capital-T Truth that gives rise to all three, but this is not shown or even implied by H. In GEB, mathematics, art, and music are all just part of the same shared web of associations, brilliantly illustrated by the mind map, a "semantic network", at the beginning of Chapter XII (Figure 70).

GEB is not a manual for building artificial intelligence. Claims of the book "falling short" of providing a complete picture or some roadmap to follow miss the point entirely. The book is much more personal than that, and more inspirational than descriptive. Hofstadter explicitly rejects his pigeonholing as a futurist, a science-fiction addict, or a technology guru. While he admits that GEB is forward-looking, these labels are all projections from the readers inspired by the book.

To me, the message of the book seems to be more like a deeply personal attempt to share a taste for something, reminding me of Antoine de Saint-Exupéry's lines about shipbuilding (in Citadelle):

One will weave the canvas; another will fell a tree by the light of his ax. Yet another will forge nails, and there will be others who observe the stars to learn how to navigate. And yet all will be as one. Building a boat isn’t about weaving canvas, forging nails, or reading the sky. It’s about giving a shared taste for the sea, by the light of which you will see nothing contradictory but rather a community of love. (QuoteInvestigator)

Woes about the complexity of the book, and much of GEB's latter-day criticism, is, I believe, mostly due to misreading that gets caught up in the constructions: a failure to see the forest for the beautiful trees. If you know what the book is really about and why it goes about its business in such a unique way, you can orient yourself accordingly, and get much more out of the experience.

On Hofstadter and the genesis of GEB

Douglas Hofstadter, born in NYC in 1945, graduated from Stanford with a Distinction in Mathematics in 1965. He then took a meandering academic route, as one does (or used to do), eventually receiving his Ph.D. in Physics ten years later, in 1975 — at 30. GEB was born during this same fruitful period of time, though Hofstadter shelved the project until he had finished his degree.

To stress the point, Hofstadter wrote the book in his late twenties and early thirties as his first foray into writing for a general audience. Once freed from the Ph.D. grind, Hofstadter rekindled his passion for Gödel and analogies, and set out to refine his dormant manuscripts and seminar materials into a complete book.

Hofstadter's academic background is in mathematics and physics, but after his Ph.D. he went on to "retool himself" as an artifical intelligence researcher. The damning Lighthill report of 1973 and the many disappointments during the decade led to the first AI winter, which would go on well into the eighties. Hofstadter entered a field in flux.

Where the mainstream AI community was moving from daydreaming to paying the bills, with a focus on applications and "expert systems" of more immediate utility, Hofstadter set out on a rather different trajectory. Building on the ideas that had excited him years before, Hofstadter set out to investigate nothing less than the fundamental cognitive processes that animate human minds. He eventually landed a position as assistant professor in Indiana. Hofstadter and his research teams have worked on computer models of mental mechanisms ever since.

In the early days of personal computing, publishing a book was no easy task, but Hofstadter, ever the visual purist, went the grueling extra mile to get the book printed exactly the way he wanted. Hofstadter finished writing GEB while teaching at Indiana University, but astonishingly somehow found the time and energy to lay out GEB himself, with help from the latest software of the day. He had access to Stanford's press through association, and with some assistance managed to shape the look and feel of GEB to his exacting standards.

Despite all the extra logistics involved in having a day job in Indiana and a side project in California, Hofstadter eventually prevailed. A veritable labour of love, years in the making, GEB was finally let loose on the world in 1979, to perplexed acclaim.

1979 : Ayatollah Khomeini returns to Iran from exile, Margaret Thatcher becomes UK prime minister, McDonald's introduces the Happy Meal, Sony Walkman launches, Michael Jackson releases "Off the Wall", Pink Floyd's releases "The Wall", smallpox certified eradicated, last natural cases of polio reported in the US, one-child policy introduced in China, first commercial spreadsheet program VisiCalc released, Compact Disc (CD) invented, worldwide per capita peak oil production, eating disorder bulimia first described and named, first Sudoku, human-powered aircraft Gossamer Albatross crosses English Channel, Kenneth Iverson wins Turing award, Douglas Adams' "The Hitchhiker's Guide to the Galaxy", Italo Calvino's "If on a winter's night a traveler", Atari "Adventure", films "Alien", "Apocalypse Now", "Moonraker", and "Being There" released.

Since GEB, Hofstadter has written several books since, around similar themes, or central idea, but I Am a Strange Loop (2007, Basic Books) is the most direct spiritual successor to GEB. Originally grafted off the preface to the 20th anniversary edition, I Am a Strange Loop is another attempt to clarify Hofstadter's views on consciousness in "a salad of metaphors and analogies".

The core of GEB

We are now ready to discuss what GEB is all about.

In the preface to the 20th anniversary edition of Gödel, Escher, Bach: an Eternal Golden Braid, Hofstadter gives the definitive account of what the book is about, and what its principal thesis is:

Hofstadter's Summary

  1. "GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter." (All emphasis mine)

  2. "What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle? What is an "I", and why [does it seem to only appear in our particular brain hardware]?" (writing open some of H's wild analogizing)

  3. "GEB [builds] up an analogy that likens inanimate molecules to meaningless symboles, and further likens self (or "I" or "soul" [, etc.]) to certain swirly, twisty, vortx-like, and meaningful patterns that arise only in particular types of systems of meaningless symbols." (Emphasis H)

Let's take a closer look at these, expanding on Hofstadter's explanation in the preface.

A very personal attempt

Gödel's proof is the starting point and the true foundation of the book. Indeed Gödel's trick is the original strange loop. However, in the process of writing GEB, Hofstadter realised that his interests in other fields outside mathematics and cognitive science were all intimately linked with the object of his study. This was particularly true of his musical and aesthetic taste, and the patterns he relished in the works of Escher and Bach, among others.

"[T]o deprive my readers of the connection that I myself felt so strongly," Hofstadter realised, "would be nothing less than perverse." There is no GEB with out E, and B, and the rest. That is, the book is fundamentally personal: it presents the main material in a reasonably accessible way and in addition includes Hofstadter's unique musings and offerings.

The book is personal in another way as well. It is true that all author's probably inadvertently write something about themselves in their works, but Hofstadter's experiences have influenced his research and writing in a particularly profound way. He grew up with a speech-impaired sister, which led him to think about human language processes early on. He also grew interested in the mind-body question at a young age, influenced by some childhood trauma. Personal tragedy features prominently particularly in Hofstadter's later works.

GEB, however, is quite an up-beat book, at least on the surface. Indeed writer Jeremy Bernstein (according to H) captured the spirit of Hofstadter's achievement by saying that the work has a "youthful vitality and wonderful brilliance." Hofstadter was, in his own words, "pretty young" at the time, but it is exactly that wild exposition and excitement of a passionate young man that makes GEB the riveting read that it can be.

GEB's success can perhaps be explained by Hofstadter's rare position at the intersection of his varied interests. He was simultaneously immersed in high culture and creative linguistic games while being sensitive to the unique beauty found in the order and function of formal systems. Young Hofstadter was both a romantic and a hard scientist.

Once H, a lifelong analogy maker, realised that the patterns he loved in different spheres of mental activity could all be shadows of the same abstract thing, it all fell into place. That thing, of course, is the strange loop, aka "tangled hierarchy".

Broadening from G to E and B falls out fairly naturally when you start from the notion of shining a light on this abstract thing. And the Carrollian dialogues follow from there. In these tangents, H finds the colour that allows him expand on his thinking in a personal way and also enables him to write a more engaging book for an educated general audience.

So GEB is a choice selection of projections of this powerful pattern; shadows of an object from a higher dimension. The cover image, a photograph by Hofstadter himself, is shadows of letters (G, E, and B) cast off of a strange cetral figure — meaningful patterns, varying with the point light source and perspective.

To be brutally concise, the main course in GEB is Gödel's theorem and Hofstadter's analogies. But in the process of digesting those, don't miss the treat of Hofstadter's unique character at play.

Out of selfless stuff, a self

When words begin to fail, something special and fundamental is at hand. Take 'soul, or 'I', or 'self'. We all have some notion of what these words point at, and that's enough for everday communication, but the concept itself remains elusive and slippery and somehow fuzzy.

And it's not just because 'self' has at best a dubious physical presence: there are many abstract and immaterial concepts that are better encapsualted by their name, where the picture of meaning is much clearer. Say 'prime', or 'hope', or 'childhood'.

Similarly, Hofstadter's 'strange loop' and its sibling terms point at an elusive concept. So much so that singular words are not enough: the loop H is talking about is strange in some way, the hierachy tangled. We reach out to analogies and attributes when there is no alternative.

Of course, philosophers have always tried to coin fundamental words and nail down ever more complicated concepts. Some would say that that is what philosophy is all about. But with GEB, Hofstadter attempts something even more audacious. He builds an analogy — a link — between two elusive concepts: the mysterious origin and nature of 'self', and Gödel's logic trick in formal systems.

Hofstadter writes: "GEB was inspired by my long-held conviction that the 'strange loop' notion holds the key to unraveling the mystery that we conscious beings call 'being' or 'consciousness'." And further on the strange loop at the heart of Gödel's theorem: "I practically heard [the secret behind the nature of selves] screaming up at me from the pages of [Gödel treatise] Nagel and Newman."

GEB is in a sense an attempt to complete the picture on both sides of the analogy: human selves are able to 'perceive themselves', but the mechanism is unclear, whereas with Gödel's trick and formal systems, the mechanism is clearly there, but its meaning or significance is limited to breaking formal systems.

Hofstadter then offers that if you think of a self-referential, strangely looping Gödelian formal system as being able to talk about itself as becoming "self-aware" in a primitive sense, then so, too, could one go about explain human self-awareness as a construction built on top of a strange loop of sorts, somewhere deep down.

Or put another way: consciousness is not a binary thing, but more like a spectrum of states, where Gödelian formal systems are towards the simple end, and humans towards the other — or at least further along on the ruler.

Interlude: Gödel's theorem in a nutshell

Russell's Principia Mathematica was built to avoid self-reference. Gödel found a way to re-introduce them through his numbering scheme. And not just for the PM version, but all systems of equal strength. A monumental result.

Based on Gödel's Theorem by Nagel and Newman (revised ed., NYU Press, 2001; foreword by one D. Hofstadter). See GEB or N&N for the details of the proof.

Gödel showed:

  1. How to construct a formula G of PM (any system of Principia Mathematica power) that represents, via Gödel numbering, the meta-mathematical statement: "Formula G is not a theorem of PM". (The strange loop)

  2. G is a theorem in PM if and only if ~G ("not-G", the opposite) is a theorem in PM: if PM is consistent, then theorem G is undecidable. (The paradox)

  3. G, while undecidable, is still a true formula, by claiming that no integer has a certain arithmetical property, which indeed is the case. (The side channel truth)

  4. Because G is both undecidable in PM, and true, PM must be incomplete. We cannot derive all arithmetical truths from axioms of PM (or any extensions). (The trap)

  5. How to construct a formula A of PM that represents the statement "PM is consistent", that A implies G, and finally that A is not a theorem in PM. Consistency of PM cannot be established within PM itself. (Incompleteness)

Meaningless symbols, meaningful patterns

Hofstadter stresses the wonder that is the fact that Gödelian self-reference in formal systems, the "skeletal self", is built from nothing but "meaningless" symbols and their "mechanical shunting". Self-reference emerges from mere symbolic data manipulation, done without concern for content or context, without added meaning or insight into the ideas embedded within. Gödel's result is astonishing, magical, and yet as solid as any mathematical truth.

Of course, as H points out, some believe that all meaning is created in human brains, and that Gödel's result is just a figment of the "semantic magic" that our brains alone are capable of generating. It's the ancient question: is the mystery of consciousness solved by "the stuff out of which brains are made, [or] the patterns that can come to exist inside the stuff of a brain."

GEB obviously advocates for the latter, but then the question is "How would those patterns work then?" It all comes down to meaning. For Hofstadter and GEB, the great gift of Gödel's work is not the incompleteness result itself, but the notion that a statement's meaning can have deep consequences beyond its immediate reach.

"A crucial part of [GEB's] argument rests on the idea that meaning cannot be kept out of formal systems when sufficiently complex isomorphisms arise." In Gödel's proof the "upside-down causality" of the key formula G is the systems downfall, according to H. The little extra meaning in the sentence makes all the difference.

Hofstadter then picks up on this mechanism by drawing an analogy between the meaningless symbols of a formal system and the inanimate molecules of a brain. What if the patterns in one could yield similar results in another medium? What if selfhood was such strange loopy patterns inside the inanimate device that is the brain?

The way this would work is that an "I" emerges when the patterns in a brain mirror the brain's mirroring of the world, including eventually themselves. As the patterns connect with their model, the patterns become "real" causal entities. Like Gödel's misbehaving formula makes the whole formal system unstable, a loop in the brain's model could bring forth wild emergent properties.

Specifically, Hofstadter presents self-awareness as a feature of any system that models its world with sufficient fidelity to include a representation of itself. In those sufficiently powerful systems, a self-mirroring strange loop of some degree is inevitable. This can be seen as a kind of liberal generalisation of Gödel's theorem.

Finally, Hofstadter posits that consciousness, or 'selfness', is measured on a continuous spectrum rather than as a binary thing. The richness of the strange loop involved is the measure. A loop in a simple formal system, no matter how strange, will not give rise to a personality. An organic brain is vastly more complex and present in the physical world, and therefore capable of much more elaborate self-reference.

Certainly there's more to life and "self" than a formal system captures. The point is in the analogy. "GEB is in essence a long proposal of strange loops as a metaphor for how selfhood originates."

A GEB Reader

To me, the key to reading GEB lies in the preface to the 20th ann. ed., where I dicovered that Hofstadter wrote GEB under the working title Gödel's Theorem and the Human Brain (GTatHB).

I'll unpack GEB's structure a little, as I see it.

The working title

From the author's excellent chapter by chapter Overview to GEB, it's easy to pick out the general shape of the book. There are two parts: GEB, and EGB. These are best understood — keeping the working title GTatHB in mind — as an introduction to Gödel's theorem, and then the construction of its extension towards the human brain, respectively. Together they form the story of a grand analogy.

Obviously all of GEB's themes are developed simultaneously in parallel throughout the book, and indeed the complete tour of Gödel's deep idea spills well into the second part, so the arc is far from fixed. However, I personally found this dichotomy to be useful, while reasoning about the book's shape. And form is content, as Hofstadter frequently reminds the reader.

As a deeply personal work, Hofstadter relishes the opportunity to showcase his experiments with literary form in GEB. Each chapter presenting the main content of the book is preceded by a conceptual introduction in the form a dialogue. It follows then that these two alternate all the way through.

Hofstadter's dialogues are inspired both by Lewis Carroll's example and the musical structures, particularly the weaving canons of titular Bach. Similarly Escher's art, liberally sprinkled throughout the book, adds flavour and intrigue to both the dialogues and the running main exposition. These are all best understood as Hofstadter's character at play, a rare high-bandwith view into the writer's brain.

A map of GEB chapter structure

Part I: GEB Part II: EGB
Preface to GEB's 20th anniversary edition Chapter X: Levels of Description, and Computer System
Introduction: A Musico-Logical Offering Chapter XI: Brains and Thoughts
Chapter I: The MU-Puzzle Chapter XII: Minds and Thoughts
Chapter II: Meaning and Form in Mathematics Chapter XIII: Bloop, Floop, and Gloop
Chapter III: Figure and Ground Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems
Chapter IV: Consistency, Completeness, and Geometry Chapter XV: Jumping out of the System
Chapter V: Recursive Structures and Processes Chapter XVI: Self-Ref and Self-Rep
Chapter VI: The Location of Meaning Chapter XVII: Church, Turing, Tarski, and Others
Chapter VII: The Propositional Calculus Chapter XVIII: Artificial Intelligence: Retrospects
Chapter VIII: Typographical Number Theory Chapter XIX: Artificial Intelligence: Prospects
Chapter IX: Mumon and Gödel Chapter XX: Strange Loops, Or Tangled Hierachies


Notable dialogues

Not all of the dialogues in GEB are equally successful, in my view, though all of them do accomplish their intended purpose: the introduction of a potentially new concept.

The one's you shouldn't skip are:


So there, my thoughts on reading GEB.

To understand Hofstadter, you basically need to know only one thing: analogies are "the fuel and fire of thinking", the secret ingredient in cognition and learning. Analogies, 'isomorphisms' in some contexts, are the map to Hofstadter's world.

GEB is, in a nutshell, an attempt to share the wonder of a particular, grand analogy that Hofstadter made when he realised what formal systems and cellular biology have in common. Hofstadter saw multitudes in Gödel's remarkable result, and to do this big idea justice, he was driven to write the most colourful exposition he possibly could. The book is a wild paean to the abstract pattern of the strange loop, and the many shadows it casts in a range of disciplines and traditions.

With GEB, Hofstadter does not give us a blueprint for AI other than by offering the reader the abstract pattern that is the strange loop, and suggesting that it could be the missing link between the inanimate and the animate: the solution to the mystery of consciousness.

EXTRA: Trivia