Reflections on ‘Gödel, Escher, Bach: an Eternal Golden Braid.’

This book has been following me for a long time. I can first recall its name cropping up around seven years ago, at one of the myriad of mathematical training camps I attended as a bright-eyed school student. Perhaps I was eavesdropping, as the critical conversation returns to me now only in fits and starts. However it happened, I became aware that several of my mathematical peers had all recently read a certain book, and that this book was called ‘Gödel, Escher, Bach’ [hereafter GEB]. They were discussing it with great fervour, and in my growing mind that three-word title immediately held an air of mystery, a golden lustre. The memory fades, of course, but what remains is a particular perception, that these friends were all bonded by some remarkable shared experience. I felt that I would never truly be a part of their circle until I had read this book too.

I was wrong in this regard, unsurprisingly: four years of the Trinity College Cambridge mathematics experience was more than enough to bond me to these folk for life. GEB returned to my mind only once, when one rainy afternoon I caught sight of the spine of an intimidating-looking copy, hiding in the mathematics corner of Trinity library (both me and the book). But there was always something more pressing to be getting on with – when is there not? – and so I didn’t read it then either.

I’m at Oxford now, working on a PhD in analytic number theory, and most nights I wear away the long age of three hours between my after-supper and bedtime by singing in one or another choir. Last year I was in The Oxford Gargoyles, a jazz a cappella group. There I met Sam Rice, an undergraduate mathematician, and Lachlan Hughes, a mature student studying German and Italian. Somehow it came up in conversation that Rice himself had read GEB, and possessed a copy he would willingly lend me, and that Lachlan – a Bach devotee – had in the past tried to read it twice, but had got stuck halfway through, on both occasions. At that time I was in the middle of a self-imposed Shakespeare reading marathon, but I took up Rice’s offer, being eager for a little diversion from the never-ending Henrys, and began my assault on GEB’s 742 extraordinary pages. After seven years, I would now finally know what all the fuss was about.

‘Gödel, Escher, Bach: an Eternal Golden Braid’ was published in 1979, and won Douglas R. Hofstadter the Pulitzer Prize for General Non-Fiction.  It has attained a cult-like status among certain communities, particularly Computer Science, with Hofstadter himself viewed as the ultimate guru. There’s something about GEB that makes everyone want to talk about it. Though it is a mighty tome itself, it would be dwarfed by a collected edition of all the reviews and articles written about it. Just have a scroll down this page, say:

or look at the sheer number of Quora posts, Reddit threads. They even ran an examinable course on it at MIT:

I am painfully aware that my own reflections are only adding to this great heap. Such a phenomenon can be partly explained by the fact that, whatever GEB actually is, that which it is is completely unique. And uniqueness is a rare and sought-after commodity, particularly from the creatively impoverished breed of writers known as reviewers. GEB oozes raw originality on every page, and, by writing a review, we can only hope that some of it might rub off.

Later on, I will come to assess whether Hofstadter and his book deserve their laudation. For now, there is a much more pressing concern. As I laid down the book beside me, just finished, coming up for air, my first thought was, “Well… what was all that about?” This question, and the lack of a satisfactory answer, plagues every existing review of GEB. And it’s not a usual question to ask, is it, of a non-fiction book? This should be your first inkling, your first warning sign, that GEB is not an ordinary book. What is GEB about? In Rice’s 20th anniversary edition, Hofstadter adds a preface, which in part explains his own intentions. In his mind, the book’s main thesis is an attempt to describe how animate thought could arise from inanimate matter. For him it’s a book about Artificial Intelligence, and it certainly contains the heady mixture of computer science, biology and philosophy than one would expect from such a theme. Since the author, the guru himself, has cleared everything up, surely he has brought an end to the 20 years of confusion? Nothing to see here.

Except of course there is. The people who failed to appreciate Hofstadter’s intended message were not stupid. For a start, the biological and computer science strands are not introduced until the book is well over 300 pages old. Lord of the Flies, The Catcher in the Rye, The Hitchhiker’s Guide to the Galaxy: all three are wound up by the 300 page mark, yet it seems Hofstadter hasn’t even started! Somehow, while developing his original theme, fleshing it out, drawing analogies, wandering off on flights of fancy, Hofstadter created such a rich body of work that in the final performance the supporting acts are completely indistinguishable from his original star attraction. He knows what he meant the book to be about: that doesn’t necessarily mean that he can speak for how the finished product ended up!

I think that the best explanation for the content of GEB comes from thinking of it as four separate books, not one. This is a slightly unsatisfactory model, as rules of three pervade Hofstadter’s magical narrative, but – as every working mathematician knows – sometimes beauty is less important than truth. Anyway, the first ‘book’ is a very well-written popular exposition of Gödel’s Incompleteness Theorem. This is a theorem of logic: perhaps the theorem of logic. The author takes plenty of time to introduce formal logical systems, and uses a couple of brilliantly-conceived toy models as explanatory tools – the MU and the p-q system. He builds up to the final dénouement with the verve of a novelist. This book most resembles an ‘ordinary’ popular science volume, and comprises most of GEB’s first act.

However, rather than satisfying himself with ‘run-of-the-mill’ pedagogy, Hofstadter is also a master of analogies, subtle and beautiful, which on some very high level echo a main idea in the proof of Gödel’s theorem. My favourite of these is the image of a record player which claims to be infinite-fidelity – it can play any record perfectly. The record player’s owner has a cheeky adversary. This adversary, after carefully studying the blueprints of the contraption, creates a record which, when reproduced perfectly, plays music at the exact resonant frequency of his enemy’s record player. When the enemy plays the record, his machine instantly shatters. Thus the claim is foiled: the appliance was not able to play every record perfectly after all.

As the record is to the record player, so is the Gödel string to a complete model of Peano Arithmetic. The central concept is that of self-reference, or a ‘Strange Loop’ as Hofstadter terms it, and in looking for other analogies he is drawn to reference many Escher prints, and some elements of work by Bach – the endlessly rising canon, most particularly. In fact, Bach fugues are later used as metaphors for the entire book, and Zen Buddhism also acquires a significant bit-part. Gleeful at the richness of his theme, this is what Hofstadter unwittingly makes into the second ‘book’. The analogies become expanded, and they acquire a new literary life of their own, quite outside their humble origins as explicatory tools.

The third ‘book’ is that book which Hofstadter originally aimed to write, when, as he reveals in the preface, the rich interwoven superstructure that would become the finished product’s hallmark was not yet conceived. This book explores how far one can go in viewing human brains as formal systems of neurons. In this viewpoint, the emergent phenomena, such as ‘consciousness’, come about by Strange Loops between different neural substrata. Hofstadter goes on to apply these ideas to the question of whether a computer could ever become intelligent, in the human sense, and whether it could ever become self-aware. In this book, biology and computer science go hand in hand, and there are excellent freestanding chapters on computer hardware, on programming languages, and on the data exchange inside each living cell, with DNA being both data and program, in an extraordinarily tangled web.

The fourth ‘book’ is very different in flavour. In between every chapter of GEB Hofstadter writes a short – though sometimes not-so-short – Dialogue, in which the characters of Achilles and the Tortoise ponder the topics that are being discussed in the book at large, and have various adventures. Other characters are slowly introduced – a Crab, an Anteater, a Sloth. They invite each other round for tea, go jogging, engage in an arms race over record players and record-player-destroying records, and, in one particularly thrilling moment, get stuck in a nested series of Escher prints whilst, in the real world, they lie in a helicopter being kidnapped. Here Hofstadter anticipates the blockbuster Inception by 35 years.  These Dialogues are wonderfully written, amusing and exciting, and are often a literary tour-de-force, particularly when Hofstadter – merely for the joy of it – writes the Dialogue to mimic a Bach fugue.

Okay, so we have four books. But now imagine that you were reading all of these books at the same time. Don’t picture having four separate volumes down on the table, and reading a chapter of each in turn, but really think about what it would mean to read them simultaneously, sentence by sentence, word by word even, with ideas from one book cropping up in another, constantly, for over 700 pages. For a start, those Dialogues are spread evenly throughout the GEB text – there are 21 of them in total – and they all contain analogies, analogies within analogies, which make ready the topics of the forthcoming chapters. It’s a bit like reciting a well-chosen Aesop fable before embarking on a heavy discussion of moral philosophy, only much more intricate, as the academia often finds its way into the dialogue, and vice versa, and all those other themes – Bach, DNA, quantum physics, Zen, logic, computer programming – get thrown along too, into the blender. Ah yes, I forgot to mention that Hofstadter’s PhD was actually in Quantum Physics…

Perhaps you are beginning to get a flavour of just how wildly rich GEB is.

As a brief example, let me describe the plot for the ‘Prelude’ ‘Ant Fugue’ – pun certainly intended – which are the two Dialogues beginning Part II of GEB. To enjoy these delights as a reader you would have to survive 274 pages, and make it through the definition of propositional calculus; although, to give some perspective, you would still be a long way off the half-way mark. The ‘Prelude’ begins with Achilles and the Tortoise having tea with the Crab, and being introduced to the Crab’s friend, the Anteater. They have brought the Crab a present, which turns out to be two records of Bach himself playing the harpsichord, created by means of ‘acoustic retrieval’, a wonderful newfangled technique, where from studying the movements of the molecules in the atmosphere one can retroactively model vibrations, simulating exactly the quivering of the air from many centuries ago. They start listening to the Prelude and, following the music on the Crab’s score – which happens to have some old Escher prints between the pages – Achilles talks about whether, in the fugue that is to come, one can ever really appreciate the intricacy of each individual line whilst also considering the effect of their combination, in the piece as a whole. There is also a strange little discussion about Fermat’s Last Theorem, though that is by the by.

In the Dialogue, the prelude ends and the fugue begins. Popping out one level, the ‘Prelude’ ends, and – after an excellent intervening chapter about computer systems – the ‘Ant Fugue’ begins. Pushing back in, the characters have discovered a peculiar illustration in between the leaves of the score, on which the word MU in written. [MU is a weird recurring element in the book, not unlike ‘Bad Wolf’ in the first series of the revamped Doctor Who – the one with Christopher Eccelston, remember? Thus I write my own submission to the canon of obscure references.]  On closer observation it seems that the word MU is actually comprised of small letters, which spell Reductionism and Holism, and under still closer observation these turn out to be written from smaller letters again, spelling Holism and Reductionism, which under much much closer observation are seen in turn to consist of the word MU once more. The characters discuss their different perceptions of the image.

Meanwhile the fugue is going on, and the characters are also commenting on particular effects that have occurred – the entrance of each voice, stretto, an organ point, etcetera. The Anteater begins discussing his recent trip to see the ant hill ‘Aunt Hillary’, and describes how he can communicate with the colony, and the colony communicate with him, even though the individual ants have no intelligence of their own. Mechanisms for such a process are posited, how different complexities of thought can coexist and intermingle, and Reductionism and Holism are once more discussed. Fermat appears again, with his famous marginal note being translated into musical parlance and placed in the mouth of a fictional Buxtehude, who claims to have composed a 24 part fugue which modulates through all 24 keys. But all the while, the text mimics the fugue which is being played in the background. Let me quote from the start:

“Achilles: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “MU”!

Crab: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “HOLISM”!

Achilles: Now hold on a minute. You must be seeing things. It’s plain as day that the message of this picture is “MU”, not “HOLISM”!

Crab: I beg your pardon, but my eyesight is extremely good. Please look again, and then tell me if the picture doesn’t say what I said it says!

Anteater: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “REDUCTIONISM”!

Crab: Now hold on a minute. You must be seeing things. It’s plain as day that the message of this picture is “HOLISM”, not “REDUCTIONISM”!

Achilles: Another deluded one! Not “HOLISM”, not “REDUCTIONISM”, but “MU” is the message of this picture, and that much is certain.

Anteater: I beg your pardon, but my eyesight is extremely clear. Please look again, and then see if the picture doesn’t say what I said it says.”

Thus the Dialogue continues with three interweaving fugal voices, meandering off into the discussion of ant hills. A few pages later we get the following, with a killer stage direction:

“Achilles:  Oh, dear! We’re getting nowhere fast. Why have you stayed so strangely silent, Mr. Tortoise? It makes me very uneasy. Surely you must somehow be capable of helping straighten out this mess?

Tortoise: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “MU”!

(Just as he says this, the fourth voice in the fugue being played enters, exactly one octave below the first entry.)

This delight lasts for 25 pages.

I feel I have already made a case for the sheer magic of this book. However, it must be said that the book falls down on some aspects where the average popular science book does not. It is certainly true that Hofstadter knows quite a lot about many different topics, and has had the rare insight that such disparate interests could be united in a coherent and enlightening way. Yet, from reading the book, I do not come away with the sense that he knows anything very comprehensively – particularly neuroscience. Chapter 11 (‘Brain and Thoughts’) rambles on for 30 pages, but its scientific underpinning could be summarised in a brief pamphlet, and even then with some padding. The notes for the MIT course point to several inaccuracies in Hofstadter’s science, whose correction puts some dents in Hofstadter’s broader theories. What’s more, the areas of genetics, computer science, and Artificial Intelligence have all developed hugely since the book’s publication. As a student in the Andrew Wiles Building at Oxford, I take particular delight that Fermat’s Last Theorem can no longer be used as the go-to example of an impossible mathematical quandary. No, I am not convinced that GEB represents the apotheosis of knowledge about any of its constituent topics. Certainly Wittgenstein would have hated GEB – “Whereof one cannot speak, thereof one must be silent.” Hofstadter is never silent.

One cannot overlook the fact that every single character mentioned in the book is male. The Dialogues in particular seem to propagate the establishment notion that intellectual coffee-time talk is the exclusive preserve of men.  To his great credit, Hofstadter dedicates a large passage of the 20th anniversary-edition preface to this terrible state of affairs, and berates his younger self – he was in his early thirties when he wrote GEB – for being so blind to this inadequacy. Fortunately the issue has a happy ending, to a degree, which stems from the translation of GEB into French. The translators suggested that ‘Mr Tortoise’ needed to become ‘Madame Tortue’, as  the word ‘tortue’ is feminine. Thus the French reading public will open GEB to find the intellectually brilliant ‘Madame Tortue’, running rings around her dimmer-witted friend Achilles.

GEB is overlong – much like this review. There is easily enough content to fill 550 pages, 600 at a push – I am not suggesting that GEB should be a short book – but there are several sections which do drag awfully, particularly some of the bio-philosophy later on, and there are themes which are introduced and never quite brought under the wing of the whole. The diversion early on into fractal graph plots from the author’s PhD thesis – by ‘early on’, I mean after 140 pages or so – is a case in point. More drastically, it is worth noting that all the references to Bach could be eschewed completely and the book would still function, be entertaining and be informative. I mention this only in passing, though; such a heinous act would rob the text of much of its unique charm.

With all its different themes, GEB does struggle with issues of unity. However, one should not overly criticise this aspect, but rather gawp in amazement at how it manages to hang together at all.  Though the writing style is very different, I am reminded a little of Bertrand Russels’ ‘History of Western Philosophy’, in which the language is so engaging, limpid and witty that the reader will gladly follow the author anywhere, even through the very deepest of intellectual thickets. Though he does waffle in the neuroscience sections, consider Hofstadter mid-flow, explaining his ‘Typographical Number Theory’:

“This completes the vocabulary with which we will express all number theoretical statements! It takes considerable practice to get the hang of expressing complicated statements of N in this notation, and conversely of figuring out the meaning of well-formed formulas. For this reason we return to the six sample sentences given at the beginning, and work out their translations into TNT.”

If that doesn’t get you manipulating formulae, I don’t know what will.

I heartily recommend that everyone should read GEB, and read all of it. However, this endorsement does come with an important proviso, a catch which I have cunningly left until the very end to mention. Many readers claim that reading GEB is a daunting undertaking, and that one needs to have significant mathematical interest and skill to get through it. I fundamentally disagree, but, alas, I do have significant mathematical interest and skill, and it is difficult for me to cast these off entirely and make a disinterested evaluation. GEB is not about maths, but many of its themes – most notably the sections on logic and computer programs – do require a certain brand of analytical thought to grasp, which is most often cultured through mathematics. In fact, the more I think about it, the more I have to admit that, in almost every respect, I was the perfect target audience for this book.  The range of my interests covers most of the broad sweep of the text, and even Escher is less of a stranger than he might be, me having drawn several studies of his work while at school – I’ve got my reproduction of ‘Drawing Hands’ in a file, somewhere or other. All this said, I strongly believe that Hofstadter’s exposition, and his delightful Dialogues, are more than ample enough for a conscientious reader, without any mathematical grounding, to have an extremely enjoyable reading experience.

GEB has fundamentally altered the way I think about so many ideas – not least the art of writing popular science – and I can well understand why my boyhood friends were so enraptured by it. Though it is a flawed work, I can think of few other long reads that have changed me so much (‘Earthly Powers’ maybe, but not ‘Augie March’). Every day I find myself drawn to using Hofstadter’s analogies, for thought, for music, for self-reference, and I have to constantly catch myself, as I realise with a shock that not everyone will get the reference.  GEB has even made me re-evaluate my belief in free will. Consider the book’s ultimate claim: any system that is capable of sufficiently powerful computation must necessarily be self-aware. Whether you agree with this or not, it is a truly startling pronouncement.  Read ‘Gödel, Escher, Bach’, and see how Hofstadter, in his own utterly inimitable way, develops his theme.  I’m glad that I finally did.

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