Most readers will be familiar with Bach’s music, perhaps have seen some of Escher’s prints, and might (just) have heard – as a distant incomprehensible mystery of higher mathematics – of Gödel’s theorem. Immediately intrigued by the punchy juxtaposition of these names in the title, they will read the subtitle. Ethernal and Golden suggest everlasting truth. Braid? Less poetically fanciful in translation (American Braid = British Plait) it conjures a triple spiral of the book’s heroes, a kind of higher conceptual protein chain whose DNA constituents are the ideas of E, G and B. Shoot to the book’s cover picture: a cleverly hollowed out shape is depicted. It casts the letters E, G, B as light is shone through it from three perpendicular directions. Is this perhaps an icon for the whole book? Shoot to the second subtitle and the tone is pretentious and pompous undercut by the suggestion of playfulness Carroll’s name always invokes.
We haven’t got past the cover but it is clear that the book is claiming for itself a message that is large, important, exciting and out of the ordinary. What is it?
In 1931 the Austrian mathematician Kurt Gödel published a long difficult mathematical paper whose techniques and results revolutionized mathematical notions of proof and philosophical discussions of formal reasoning. The main theorem of this remarkable work demonstrated the existence of true statements about whole numbers expressible in the language of a simple arithmetical system that could never be proved within the system. Put another way: all the truths of elementary arithmetic can never be obtained as consequences from a single list – finite or finitely specifiable – of axioms for whole numbers. And that, however many of these truths were listed as fresh axioms, there would always remain others that lay outside the net of possible proofs available in the system. In the same paper Gödel also indicated how the consistency of elementary arithmetic itself – its freedom from contradictions – could not be established by reasoning expressible within the system.
These results of Gödel, the manner of their proof, and the abstractions they rest on – as familiar to mathematical logicians as Einstein’s work is to physicists – are what the book stalks, exposes, celebrates, explains, and en-goldens.
But all is not so simple. There is Escher. There is Bach. And as background noise (or should it be foreground silence?) there is the baffle of Zen.
Escher’s startling images: hands drawing each other, mosaics where black figures on white ground are also ground for white figures, perpetual self-feeding waterfalls and self-reflecting puddles. As a bogeyman chopping at the conventions of two-dimensional representation Escher is the perfect source of disorientation, paradox, and contradiction that the author needs to jolt and re-educate the reader into the subtle self-referring distinctions of mathematical logic. Escher’s prints, like Zen aphorisms, offer instant baffling confirmation that paradox is the night side of logic: if you want one, you must face the other.
Bach’s presence throughout the opus is far less convincing. He appears as a thinned out figure preoccupied by repetition, canonical tricks, parallel constructions, fugal reversals, and infinitely ascending loops. True, Bach was fascinated by such things, but it is a small bare box to put him and his music into. And the author’s central conceit – his book as Idea from which the light of Gödel, Escher, and Bach stream endlessly – seems forced, unedifying and historically vapid.
But in the 750 pages of agile, inventive, clever writing there is much on offer. The intellectual constituents of Gödel’s work – the elements of propositional and predicate logic, formal systems of arithmetic, recursion, self-referring sentences, meta-level constructions and their codings – are all explained and uncovered with great energy, enthusiasm and unusual expository brilliance. Analogies and parallels from neurology, artificial intelligence, molecular biology, and the mechanisms used by nature to get seeds to make plants which have seeds which make……all jump and dance about in the Gödel circus.
There are clowns too. Each of the twenty chapters is prefaced by a dialogue, exchange, or tricksy debate featuring Achilles and Tortoise (by courtesy of Zeno and his paradoxes), or the Ant and Anteater, or Crab with his sideways intellect. Ringmaster Hofstadter makes sure his clowns have Cultural Depth. Their pieces are – he tells us – modelled on Bachian modes and have the names to prove it: Two Part Invention, Sonata for Unaccompanied Achilles, Contracrostipunctus (in which “explicit references to the Art of the Fugue are made. The dialogue itself conceals some acrostic tricks”), Little Harmonic Labyrinth, Canon by Intervallic Augmentation……Six Part Ricercar. The function of the clowns – presumably – is to flex and tease the reader’s mind for the next bout of exposition. Thus the Little Harmonic Labyrinth – a preface to the discussion of recursive processes – has fun with infinity on a Ferris wheel: there are genies who grant wishes, Achilles who thinks everybody in the Arabian Nights is dopey not to have made a meta-wish – a wish for more wishes, meta-genies, meta-meta-genies…..all the infinite way to GOD who, according to the genie, is “not some ultimate djinn. GOD is a recursive acronym” and so on, and so on.
Good clean mathematical fun and games easily playable, with infinitely many variations, by anybody familiar with Cantor’s theory of transfinite ordinals.
How seriously should one take all this splashing about in the shallows of early twentieth century mathematical logic? If one leaves Achilles and his pals to clown through their Bachian routines, and looks instead at the chapter summaries, such phrases (not untypical) as “implications for the philosophy of mathematics are gone into with some care: Chapter XIV) suggests that beyond the punning and music making there is serious philosophical intent. But the goods are not delivered. Time after time elementary insights, the common property of any student of mathematical philosophy, are dressed up and marched forward as Important Findings or Fundamental Questions. To take a single but central example: the whole discussion of logic which elaborates the meaning of “and”, “or”, “not”, and so on, is reduced to an insular and banally oversimple account by the complete omission of Brouwer’s Intuitionism.
Besides its intellectual superficiality the book is impoverished in other more interesting ways. Like so much of the type of sci-fi literature it closely resembles it is humanistically and culturally thin. If the book has a central topic it is that of self-reference and reflexivity. And surely the interesting end of self-reference lies not in the details of self-reproducing mechanisms, but in the phenomenon of self-consciousness. The author seems not to have visited even the banks of that great river of ideas on the self that starts with Montaigne, passes through Nietzche, and floods into Freud and Proust.
But let Mumon, the author’s revered Zen Master, have the last (paraphrased) word:
Has the book innerworth?
This is the most serious question of all.
If you say yes or no,
You lose your own innerworth.