Most readers will be familiar with Bach’s music, will perhaps have seen some of Escher’s prints, and may have heard – as a mystery of higher mathematics – of Gödel’s theorem. Intrigued by the juxtaposition of these names in the title of this book, they will seek an explanation in the subtitle: “Eternal” and “Golden” suggest everlasting truth. “Braid”, meaning “plait”, though less fanciful in American than in English, conjures up a triple spiral of the eponymous heroes of Gödel, Escher, Bach, a conceptual protein chain whose DNA constituents are the ideas of E, G and B themselves. The book’s cover picture shows a single cleverly hollowed-out shape that casts shadows of the letters E, G, B as light is shone through it from three different directions. The second subtitle, “A metaphorical fugue on minds and machines: suggests a pretentiousness which is then undercut by “the spirit of Lewis Carroll”. So much for the cover. It is clear that the book is claiming for itself a message that is large, exciting and out of the ordinary. What is this message?
In 1931 the Austrian mathematician Kurt Gödel published a long and 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 Gödel showed that, however many of these 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 he 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’s, the manner of their proof, and the abstraction they rest on, as familiar to mathematical logicians as Einstein’s work is to physicists, are what the book stalks and celebrates. But then there is Escher. And there is Bach. And as background noise (or should it be foreground silence?) there is the baffle of Zen:
Has a dog Buddha-nature?
This is the most serious question of all
If you say yes or no
You lose your own Buddha-nature.
Escher’s startling images are familiar: 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. He subverts the conventions of two-dimensional representation and his pictures are the perfect source of disorientation that Hofstadter needs to jolt 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 in Gödel, Escher, Bach 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 his music into. And the author’s central conceit as represented in the cover picture – his book as Idea from which the light of Gödel, Escher and Bach streams endlessly – seems forced, unedifying and historically vapid.
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 plants etc.) all dance about gaily in Hofstadter’s three-ring circus. There are clowns there too: each chapter is prefaced by a dialogue, exchange, or tricksy debate featuring Achilles and the tortoise or the Ant and the Anteater, or the Crab with his sideways intellect. They have titles like “Two Part Invention” “Sonata for Unaccompanied Achilles” and “Contracrostipunctus” (in which “explicit references to the Art of the Fugue are made. The dialogue itself conceals some acrostic trick”). Here Hofstadter gets silly: in “Little Harmonic Labyrinth” Archilles thinks everybody in the Arabian Nights, given the option by the genie of the lamp, 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. For God, according to the genie, is “not some ultimate djinn. God is a recursive acronym”.
These games are playable, with infinitely many variations, by anyone who is familiar with Cantor’s theory of transfinite ordinals. The chapter summaries suggest, in such phrases as “implications for the philosophy of mathematics are gone into with some care” that there is serious philosophical intent in their presentation here. But 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 central example: the whole discussion of logic which elaborates the meaning of “and”, “or”, “not”, and so on, is made insular and simplistic by the omission of any reference to Brouwer’s Intuitionism.
Besides being intellectually superficial the book is impoverished in other, albeit more interesting, ways. Like so much of the type of sci-fi literature it closely resembles it is inhumane and culturally thin. If there is a central topic it is that 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 concerning the self with its source in Montaigne, that flows through Nietszche, and reaches full flood with Freud and Proust.
Is Gödel, Escher, Bach worth reading? Has a dog Buddha-nature? No.