Putting the “B” Bach in GEB
Gödel, Escher, Bach by Douglas Hofstadter turns 40 this year. Bach’s music never had a justified place in the book – but could it find one?
By Ilari Kaila
Achilles: Frankly, I’m a little confused by the title. After all, what do Copper, Silver, and Gold have to do with each other? … Now if the title were, say, Giraffes, Silver, Gold, or Copper, Elephants, Gold, why, I could see it…
Tortoise: Perhaps you would prefer Copper, Silver, Baboons?
Achilles: Oh, absolutely! But that original title is a loser. No one would understand it.
Tortoise: I’ll tell my friend. He’ll be delighted to have a catchier title (as will his publisher).
(From Gödel, Escher, Bach: An Eternal Golden Braid (1979) by Douglas Hofstadter)
Twenty years ago, in the preface to the 20th-anniversary edition of his classic book, Douglas Hofstadter marvelled at how misunderstood its thesis has been. A treatise on the nature of consciousness, it is often wildly misconstrued as an exploration of how ‘math, art, and music are really all the same’. But one likely source of the confusion is in the name – which is, at the same time, a big reason for the book’s lasting popularity: Gödel, Escher, Bach: An Eternal Golden Braid, or GEB for short, sounds like a promise of just such a dazzling, cosmic counterpoint. Another likely culprit is Hofstadter’s own musings about music. While M C Escher’s artwork elegantly (and literally) illustrates many of the book’s themes, Hofstadter’s attempts at justifying the inclusion of Bach are mostly banal and often badly off the mark.
There are good reasons for GEB’s fame besides the sexy and marketable title, though. In its attempt to build a grand theory of minds and meanings, the book discusses an eclectic range of topics and, at its best, does it in a genuinely enlightening way. It is obviously an inspired work, even if the fundamental case it sets out to make falls flat. Like most attempts at ‘explaining’ consciousness, GEB is rooted in a category mistake: it treats our phenomenological core as just another phenomenon, making the book an 800-page exercise in begging the question. But it’s a stimulating 800 pages, riffing on fractals, Zen koans, computer languages, quantum physics and much more. To his credit, Hofstadter at least senses that the volume of a phonebook is required if you claim to be adding something to this perennial conversation.
At the centre of GEB’s thesis is a concept that Hofstadter calls the ‘strange loop’, a system of tangled hierarchies, often self-referential and paradoxical – and one that, in his view, gives rise to our sense of selfhood. Like with any complex system of logically governed symbols, his argument goes, the symbol-manipulation of our brains leads (as the 20th-century mathematician Kurt Gödel showed) inevitably to self-reference. This is what self-awareness is, in Hofstadter’s final analysis: the ability of our internal, neurally processed formal system to reference itself, to ‘talk about itself’, as a property that is mathematically predetermined to emerge.
An important strand in GEB’s exploration of such strange and loopy entities is recursion. A simple and intuitive illustration of the concept, and one that the book starts out with, is of a story inside a story: a self-similar structure, in which each new hierarchical level is nested within a lower one. This recursive nesting can either continue indefinitely (as in fractal geometry) or until the levels ‘bottom out’ (as with the leaves of a fern, or the 1,001 tales of Arabian Nights). Nature and human artifacts abound with examples of recursion and self-similarity, and Hofstadter exploits a great many to explain the concept. With his third titular character, however, he struggles.
In an effort to keep Bach relevant to the discussion, Hofstadter takes an oddly specific example (‘the gigue from the French Suite No 5’) and goes on to give descriptions that are so general they could apply to any number of tonal pieces. His main point is that the way the music moves through different keys – a process known as modulation – is recursive. But key changes are not organised in anything resembling stacks in computer languages, as Hofstadter would have it, with each new level nested within the previous one; nor do they create the expectation of ‘returning back in a reverse order’. Modulation is a defining characteristic of tonal music: it is not unique to Bach, nor is it a recursive phenomenon.
Ferns and fractals: recursion in nature and in the abstract. (Mandelbrot set animation created by Wolfgang Beyer with the program Ultra Fractal.)
Music is full of recursion, though. Schenkerian analysis – one of the dominant theoretical frameworks for tonal music (since before GEB was published) – deals with hierarchical harmonic structures that follow similar principles from the deepest levels to the surface. Other musical parameters, such as rhythm, exhibit recursion too. It would be easy to come up with any number of better examples than key changes in a random gigue. On the other hand, as GEB takes pains to demonstrate, recursion is everywhere: in plants, computer code, quantum particles, human languages. So whence the ‘B’?
More than anything, Bach’s name in the title represents a missed opportunity: his music could, in fact, beautifully serve the role that Hofstadter wanted to assign it. Beyond its relevance to recursion, it can help us think about the book’s biggest underlying subject matter, that of meaning – though it might lead us to a very different perspective from the one Hofstadter intended.