Book Review: GÖDEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID,
by Douglas Hofstadter

This is a monumental book which nonetheless succeeds in holding the interest. This is due partly to a direct and vivid style, and partly because of the dialogues which come at the end of each chapter, which the author uses loosely as a way of both relaxing the literary form and introducing new ideas to the reader.

The dust-cover reads: 'Linking the music of J S Bach, the graphic art of Escher and the mathematical theorems of Gödel, as well as ideas drawn from logic, biology, psychology, physics and linguistics, Douglas Hofstadter illuminates one of the greatest mysteries of modern science: the nature of human thought processes.'

This may be true, but it is not really the aim of the author. The central thesis of the book is that our own intelligence is in the end reducible to physical processes, and that the goal of creating an artificial intelligence is therefore possible. In the preface Hofstadter calls this 'a statement of his religion'.

He is apparently content not to prove his case; he does devote some time to try to show that the arguments against are not conclusive, in particular those derived from the existence of Gödel's Theorem. However the style is generally metaphorical rather than rigorous, and the book thereby gains in attractiveness for the general reader. The two more rigorous arguments employed are analysed below.

Because this method implies constantly drawing parallels with other fields, it leads the author to religion and philosophy. Certain writings from Zen Buddhism are put to use over several chapters, in a good-humoured way, to illustrate that words cannot totally capture the notion of truth (a strange parallel with Gödel). Towards the second half of the book Hofstadter writes against what he calls 'soulists', those who believe in a soul.

This seems necessary to the thesis of the book, because if the soul exists, our intelligence is not mechanical and therefore not programmable. The author appears to confuse human personality with consciousness: the former will appear in machines (computers) when they are capable of self-consciousness. The book thus acquires an atheistic flavour at intervals from about Ch.10, and which is dominant after Ch.15.

Structure: The book is in two halves, of 9 and 11 chapters respectively. In the first part (GEB) Hofstadter presents the key ideas of the book. In the second part (EGB) he uses them, first working towards Gödel's Theorem, (which is worked out in chapter 14) and then trying to show that this does not mean that artificial intelligence is impossible, even though it is a very long way off.

Central parts worth reading on their own:

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  Ch.1, Introduction, and a potted history of Gödel's Theorem
  
• 
  Ch.3's dialogue, 'Contracrostipunctus' (a metaphor on Gödel)
  
• 
  Ch.7, 8, 9: Typographical Number Theory (the key of the book)
  
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  Ch.13, 14, 15: Gödel's Theorem derived (for TNT) and discussed
  

On reading the above one would have a useful understanding of Gödel's Theorem; the working out of it is impressive, perhaps the more so as the author is trying to overcome it in the remainder of the book, which is his profession of faith in the future of artificial intelligence research.

The dialogues are worth reading for their witty exposition of interesting problems. Of course it has to be remembered that these dialogues are metaphor or fantasy. This applies especially in the important dialogues in Ch.10 and 20, which can be seen as statements of his beliefs. Perhaps the Crab Canon (Ch.7) is the most successful.

Some specific problems:

In Ch.15 (p.476), Hofstadter (against Lucas) argues that because Gödel's Theorem can be (recursively) added as an axiom to a system, computers can have as much 'power' as a human, although being incomplete. This argument fails for the reason given by Penrose in his later work, 'The Emperor's New Mind', which is that Gödel's Theorem itself is the discovery of a man, whereas a computer by its very nature could not have found it. And it was a result, not an axiom.

In Ch.15 (p.477), Hofstadter draws a parallel between Gödel's Theorem ('This truth is not a theorem') and the Epimenides Paradox ('This sentence is false'), arguing that the languages we use are also vulnerable to 'word tricks'; ergo, we are no better than computers. However he correctly undermines this in Ch.17 (p.579 - 581), the true parallel to the Epimenides Paradox is not Gödel's Theorem, but Tarski's Theorem ('This theorem is false') - which, being also a concrete statement about numbers, can be proven not to exist as it would then be both true and false at once. Two morals can be drawn from this (to start with):

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  a) Truth and theoremhood are different.
  
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  b) Gödel's Theorem is not a mere 'word trick', but a concrete statement about numbers as well. Thus it cannot be side-stepped.