It is not very often that a book about mathematics goes straight to the top of the bestseller charts. A publisher’s axiom is that, for every equation written in a book they are drastically reducing its readership. However, Godel Escher Bach is a wonderful exception to this theory, explaining complex mathematical ideas to a lay audience, without compromising on content. It is a work of art as well as a serious non-fiction book, drawing together similar themes in music, mathematics and art. At 770 pages long, it is not easy to summarise, but I will have a go at dealing with some of the key concepts.

Godel Escher Bach is about artificial intelligence – the idea that, with advances in technology, one day the computer will be able to think for itself. It will have its own personality and be able to escape the trammels laid down for it by its programmer. There is an argument that this is impossible – described eloquently by Roger Penrose in The Emperor’s New Mind. There is also an argument that, given enough processing power and enough complexity, that intelligence will arise automatically. This may seem more likely, after all our brains have evolved with a finite number of neurons and neural pathways. Equally, advances in computing power, which is growing at a phenomenal rate, may one day reach and even surpass our own intelligence. Then again, this could be an impossibility.

Hofstadter is definitely in the “AI is possible” camp. Written in 1979, Godel Escher Bach (GEB) is Hofstadter’s pioneering and brilliant attempt at popularising his own approach to artificial intelligence. It explores things from the ground up – How can we simulate thoughts on a computer? What process might be going on as we think?

To explore these questions, Hofstadter explores three at first apparently unrelated fields – Bach’s fugues, Escher’s prints and Godel’s Incompleteness Theorem.

Bach

Have a listen to The Musical Offering by J.S. Bach – it sounds mathematical, clinical and incredibly ornate – as if four instruments are playing, even though it is actually a solo piece. The reason for this is the structure of much of his music, which constantly refers to its own melody by inverting itself, harmonising with another copy of the melody which is staggered in pitch or in time, or even with a backward copy of itself. Hofstadter points out that the music is operating on itself. This is a key concept in the book – a “Strange Loop”.

Escher

Similar self-reference can be found in many of Escher’s works, such as “Drawing Hands” – where a hand is drawing a picture on a piece of paper, except the drawing itself is a hand, drawing a picture of a hand, which is coming out of the paper and drawing . . ..

Godel’s Incompleteness Theorem

In 1913, the philosopher Bertrand Russell produced a uniquely ambitious project, the *Principia Mathematica*, which attempted to get rid of all this self-referential nonsense in mathematics and put it on a firm, logical footing.

Godel made the startling proof that this was impossible. In any given list of numbers, if you take the leftmost number and simply add 1 on to it, then take the second left number from the next number in the sequence, and so on . . . you are left with a number, which, because it has been operated on by every number in the sequence, it cannot itself be part of the sequence.

Furthermore, any number can be represented by a corresponding computer program, for what is termed a Universal Turing Machine (a prototype, imaginary computer proposed by the British scientist Alan Turing).

A Universal Turing Machine can process any instruction on card [bear in mind that Turing was writing in the 1950s, when punched cards were a state of the art storage system], which can be defined in terms of a binary, or decimal number. Hofstadter shows that any Turing machine powerful enough to be “universal”, that is be able to run any such programme, must also be able to run a programme, which would in itself destroy the machine, since that can also be encoded as a programme.

Therefore, it is impossible to devise a way of encoding all mathematics, as “strange loops” are unavoidable. This is also relevant also to biology, since, in the final chapters of the book, Hofstadter thinks of DNA as a computer program, and indeed there are self-replicating strands of DNA, encapsulated in proteins which hijack cells in order to reproduce themselves, and so infect their host. We call them viruses.

To make all this more digestible, Hofstadter introduces each section with snippets from a dialogue between Achilles and the Tortoise, in imitation of Lewis Carroll. While he doesn’t quite manage the beauty and intrinsic childishness of Alice in Wonderland, nevertheless this is a highly original and beautiful approach to mathematics which anyone can understand, even if we are not all able to follow every twist and turn of this fantastic book.

I read GEB first when I was in my late teens and have returned to it occasionally ever since. Don’t get the “20th Anniversary Edition”, however. Try to find an earlier copy since the later edition is very cheaply produced in order to keep costs down and this does not do justice to the many Escher works included in the volume. If you can afford a hardback copy, so much the better. This is one book you will never tire of reading.