GODEL, ESCHER, BACH: An Eternal Golden Braid
Vintage Books, New York, 1979
777 pp. Paper, $8.95
Suggested age level: 16 to adult
Reviewed by T. G. Cleaver
"Godel, Escher, Bach" is Douglas Hofstadter's Pulitzer Prize winning
book. It's hard to describe what the book is about. "The philosophy of
science" comes as close to a simple description as I can come, but the book is
much more.
What about that peculiar title? Hofstadter tries to tie together the
works of the mathematician Godel, the artist Escher, and the musician Bach.
And it makes a kind of weird sense.
M. C. Escher is the artist who does those black-and-white lithographs
that trick the eye. Perhaps you remember that picture of people walking up
and down stairs, but you can't tell which way is up. Another famous one is a
picture of two hands, each hand holding a pencil and drawing the other hand.
J. S. Bach was a master of the fugue, a musical form that capitalizes on
playing a theme (melody, counterpoint), then replaying it transposed,
upside down, and backwards. The result can be that the music appears to "eat
its tail", and also that while it appears to be going up and up, it ends up
where it started. Hofstadter points this out as an auditory equivalent of
what Escher does with graphics.
Kurt Godel was a mathematician who proved that we can't know everything
for sure. What he actually said was, "All consistent axiomatic formulations
of number theory include undecidable propositions." This is know as Godel's
Incompleteness Theorem, and its implications are enormous. One conclusion
that can be drawn from this is that although a mathematical statement may be
true, it may not be possible to prove that it is true.
Hofstadter gives us Russell's Paradox: Define Run-of-the-mill sets as
sets which do not contain themselves as members. Define Self-swallowing sets
as those that do. Now define a specific Run-of-the-mill set which is the set
of all Run-of-the-mill sets. Is it a Run-of-the-mill set or a Self-swallowing
set?
Hofstadter is fascinated with these paradoxes, so he goes on about them
for a whole book. It is full of mathematical, artistic, mathematical, and
linguistic paradoxes, all presented in a witty, informal style. The book is
very worthwhile reading, if only for its insights on the nature of truth and
the nature of proof.