Hofstadter's great achievement in Gödel, Escher,
Bach was making abstruse mathematical topics (like undecidability,
recursion, and 'strange loops') accessible and remarkably entertaining.
Borrowing a page from Lewis Carroll (who might well have been a
fan of this book), each chapter presents dialogue between the Tortoise
and Achilles, as well as other characters who dramatize concepts
discussed later in more detail. Allusions to Bach's music (centering
on his Musical Offering) and Escher's continually paradoxical artwork
are plentiful here. This more approachable material lets the author
delve into serious number theory (concentrating on the ramifications
of Gödel's Theorem of Incompleteness) while stopping along
the way to ponder the work of a host of other mathematicians, artists,
and thinkers.
The world has moved on since 1979, of course. The
book predicted that computers probably won't ever beat humans in
chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl
record, which serves for some of Hofstadter's best analogies, is
now left to collectors. Sections on recursion and the graphs of
certain functions from physics look tantalizing, like the fractals
of recent chaos theory. And AI has moved on, of course, with mixed
results. Yet Gödel, Escher, Bach remains a remarkable achievement.
Its intellectual range and ability to let us visualize difficult
mathematical concepts help make it one of this century's best for
anyone who's interested in computers and their potential for real
intelligence. --Richard Dragan.
Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel:
biographical information and work, artificial intelligence (AI)
history and theories, strange loops and tangled hierarchies, formal
and informal systems, number theory, form in mathematics, figure
and ground, consistency, completeness, Euclidean and non-Euclidean
geometry, recursive structures, theories of meaning, propositional
calculus, typographical number theory, Zen and mathematics, levels
of description and computers; theory of mind: neurons, minds and
thoughts; undecidability; self-reference and self-representation;
Turing test for machine intelligence. |
