Gödel, Escher, Bach: An Eternal Golden Braid - Douglas Hofstadter

If you were to be marooned on a deserted island, and could bring three books, which ones would you choose? This familiar question, suitable to get a dinner table conversation going, or to serve as an icebreaker among strangers, deserves careful consideration. Who knows, one day you might be forced to leave your home or your country in a hurry, with just a minute to decide what to bring...

As for me, I would make sure to choose some books which would last for a long time, and which I had anyway saved up for a more careful reading once I retired and had more time for reading. (Alas, now that I have been retired for four years, it turns out that I do not have the time and patience needed to fulfil my youthful resolve to broaden my education, which also included learning Chinese and Arabic.) Upon careful consideration, here is my selection:

  1. "The Complete Works of William Shakespeare" (Abbey Library, 1100 pages in small print)
  2. A chess book. I have not been able to decide between my old favorite "Schachgenie Aljechin" by H. Müller and Pawelczak, and "Fundamental Chess Endings" by K. Müller and Lamprecht, or perhaps "My 60 Memorable Games" by Bobby Fischer or "My Great Predecessors" by Kasparov (I have not read the two last-mentioned).
  3. "Gödel, Escher, Bach" by Hofstadter (Penguin Books, 777 pages).

None of these books is what you might call a page-turner. But they all offer a good deal of resistance, and so invite re-visiting and re-reading.

"Gödel, Escher, Bach" struck me as a bombshell in 1980. It addressed many of the subjects that had fascinated me for a long time: the foundation of mathematics, symbolic logic, the nature of language and meaning, artificial intelligence, and the origin of life. And it raised as many questions as it answered.

The book spans over many fields, from Zen Buddhism to the art of Escher and Magritte and to DNA molecules. The main text is interspersed with playful dialogues between "Achilles" and "the Tortoise", offering an offbeat, whimsical perspective of the more serious discussion. The latter is presented in accordance with Einstein's dictum: "Make everything as simple as possible, but not simpler." A mathematical background is not required, although it may make the reading a little easier.

A self-referential Escher drawing.

The book sparkles with ideas and associations. It seeks to tie together the works of Gödel, Escher and Bach, but it is clear that it deals mainly with Gödel. Escher and Bach are introduced chiefly to offer examples of the unifying concept of self-reference, and to lighten up the text with some interesting discussions and illustrations which provide variations of the main theme in a different context

Out of the great achievements of 20th century science, that of Gödel is probably the least known and understood by the scientific community, relative to its fundamental importance, to say nothing of the general public. You might say that Gödel's main work, published in 1931, completed the great earthquake that happened in science with the establishment of the Special and General Theories of Relativity and of Quantum Theory. It overturned the foundations, not only of physics, but of mathematics itself.

Ever since Euclid, it had been accepted that mathematics dealt with deriving the logical consequences of a few simple postulates: axioms, or "self-evident truths". Euclid's own work "Elementa" was exactly that: a collection of truths about geometry derived from a few axioms. Much later, in the 17th century, the Calculus was developed by Leibniz and Newton. It involved the calculation of areas and volumes through successive approximations, leading to infinite series. The methods worked, but caused a great deal of confusion due to the underlying concept of adding an infinite number of "infinitely small" increments to arrive at the result. During the 19th century, a more solid foundation was laid by mathematicians such as Cauchy and Weierstrass.

A low point in my own education in mathematics came in high school, when I had missed a few lessons due to illness. I was unable to grasp, or at least to apply, Weierstrass' famous definition of continuity (and I failed miserably in a math test for the first and last time): "f(x) is continuous at ξ if for every positive number ε, no matter how small, there can be determined another positive number δ = δ(ε) such that |f(x) - f(ξ)| < ε for all points x for which |x - ξ| < δ." Today this seems pretty obvious to me, but I can see why this would be a stumbling stone for a high school student!

A further attempt was made to anchor arithmetic in basic axioms of logic. This was a rather esoteric exercise. "If a implies b, and b implies c, can we be certain that a implies c, or do we need a further axiom If a implies b and b implies c, then a implies c"? Are proofs involving induction valid? Many similar issues, some perhaps more contentious than these, had to be settled. This work culminated in Russell's and Whitehead's Principia Mathematica in 1910-1913, where in part II, page 362, we find:

From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2.

In the course of his investigations, Russell discovered this paradox: "Consider the set of all sets that are not members of themselves. If such a set exists, it will be a member of itself if, and only if, it is not a member of itself." - This paradox echoes the famous paradox of the Cretan Epimenides: "All Cretans are liars" (to be understood that all Cretans lie all the time) and points forward to Gödel's results.

One may ask why mathematicians are so obsessed with paradoxes, which may seem contrived, involving self-references at different levels of abstraction: "The following sentence is true. The previous sentence is false." etc. The answer, pointed out again and again in Hofstadter's book, is that self-reference is a key feature of intelligence, and of life itself.

Kurt Gödel 1906 - 1978.

Gödel's incompleteness theorem tells us that all consistent axiomatic formulations of number theory (including Russell's Principia Mathematica) include undecidable propositions, i. e. propositions whose truth or falsehood cannot be decided within the theory. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set. - I should point out that this does not prove that there are any "unprovable truths", just that there are truths which cannot be derived within the axiomatic formulation of number theory.

Gödel's proof rests on the observation that any sentence consisting of a string of letters or symbols can be assigned a unique natural number by using prime numbers. The Gödel number thus generated allows us to retrieve a sentence through factorisation. There is a one-to-one correspondence between a sentence and its Gödel number.

Now, as each sentence within the axiomatic system corresponds to a natural number, the rules of arithmetic can be applied to show that within the system a sentence referring to itself can be formed, the truth of which cannot be determined within the system. Moreover, it can be shown that the consistency of the axiomatic system cannot be demonstrated within the system itself, so any axiomatic system on which number theory is based will necessarily be incomplete, and it will be impossible to prove its consistency within the system itself.

So, what does it all mean?

The crucial point is that Gödel's discoveries concern self-referential systems, and such systems are much more than a mathematical curiosity. For instance the DNA molecule contains a coded message with instructions on how to produce descendants of itself in a very roundabout way (using proteins and enzymes to generate cells and organs and bodies and gametes). Likewise, the Holy Grail of Artificial Intelligence is the development of computer programs that modify themselves in response to their environment. And the human brain itself is extremely self-referential, on several levels. Just think of what goes on in your brain, when you think of yourself, or when you think about thinking of yourself! Hofstadter believes that this process underlies the experience of consciousness, meaning and self; and that complex systems built upon simple mechanistic components, such as neurons and transistors, can evolve awareness and fundamentally unpredictable behavior.

Douglas Hofstadter (b. 1945)

This raises any number of interesting questions. For instance: The human brain contains a hundred to a thousand billion neurons. This is a number only moderately larger than the number of humans on this planet, some 7 billion. Suppose we ran a simulation of a brain, using one person per neuron, who would be in touch with perhaps a thousand other "neurons" and send signals to other "neurons" in response to inputs according to mechanistic rules - of course under the assumption that we know what those rules are, and that the structure and detailed wiring of the brain, at a given moment, had been mapped. (Yes, I realize that that is quite an assumption!) Such an assembly should in principle be capable of simulating a brain, even if at much lower speeds. Could such an entity "experience" the world? Could it become conscious? - I suspect that one would have to add sensory impressions, or one might be stuck with a catatonic being, a "cabbage case". But does the substrate matter of the brain itself matter: neurons, transistors, or persons simulating neurons?

In "The Emperor's New Mind", R. Penrose argues that it does. Computers are restricted to algorithmic processing, while humans are able to achieve insights through non-verbal thinking and an intuitive sense of Truth and Beauty. Even a child can see that the conscious mind cannot work like a computer, no matter how complex. (Hence the title of his book.) - Personally, I find Penrose's arguments unpersuasive, but of course that does not mean much ... The debate rages on with undiminished intensity today, 25 years later.

The fascinating question of what a mind is, is further explored in "The Mind's I" by Hofstadter and Dennett.

Further reading

  1. A concise essay on Gödel's findings.
  2. A discussion of some common misconceptions about Gödel's results. (A collection of links.)
  3. A Gödel biography, with some technical information on his work. (Stanford Plato archive.)

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    Last edited or checked June 6, 2013.  

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