Douglas R. Hofstadter is College Professor of Cognitive Science and Computer Science; Adjunct Professor of History and Philosophy of Science, Philosophy, Comparative Literature, and Psychology at Indiana University. (For his books I've read, click on his name.)

He won the 1980 Pulitzer Prize for general non-fiction for this book, aka GEB for short. GEB has inspired thousands of students to begin careers in computing and artificial intellegence. A sub-subtitle of GEB is “A Metaphorical Fugue on Minds and Machines in the Spirit of Lewis Carroll.” I’ve now read it three times – first a quick read in 1981 to get the general ideas, then in 1994 to follow carefully Gödel’s mathematical logic, and in 2006 to focus on the so-called “strange loops” that connect the mathematical patterns and similarities of the logic of Gödel, the art of Escher, and the fugues of Bach. Since I’ve always had a stronger geometric and logical intuition than a musical one, the math of Bach's music came to me last. I have always been a great fan of Escher’s work (owning four Escher books), so my natural starting place was with the art and the logic.

A special feature of GEB is that each chapter begins with “Dialogues” (such as Zeno used in ancient Greece) of characters: the Tortoise, Achilles, and the Crab, whose conversations illustrate subtle points. If you are familiar with Escher, you can appreciate some of the strange things he did in some of his drawings (often breaking the laws of projection of 3D into 2D), and such as drawing different parts of a scene from different perspectives to make an “impossible whole.” Bach used “math tricks” such as inversion, reflection and “sliding scales” to compose, often by improvising, some amazing music. When Hofstadter shows how similar these Bach-Escher tricks are, it often “blows your mind” and lets you hear what you see!

Douglas R. Hofstadter builds up the logic slowly, using only rational numbers (non-negative integers and ratios of such), so you can follow his development of a formal logical system of axioms and theorems based upon the axioms. He constructs a formal logical system that is realistic enough (which requires the genius of the so-called Gödel numbering) to prove Kurt Gödel’s famous “Incompleteness” (or “Formally Undecidable”) Theorems. Simply put, “Incompleteness” means you can neither prove nor disprove some theorems using axioms of the formal system. This may sound innocuous, but it means that there are math truths that you know are true but you can’t formally prove that they are true or false. They are connected with “self-referential” concepts. A well-known example (from ancient Greece) is “This statement if false.” One must “go outside the box” of the formal system to see some truths. Similar to self-referential statements is self-replication, which brings in computing ideas like a “Turing Machine” and “Artificial Intelligence” (AI). Hofstadter didn’t dwell enough on the fact that he designed his formal logical system with a structure that mirrors that of DNA. For “algorithmic” systems (like formal logical systems, Turing machines or AI), this means that there are truths that lie “outside the box” (outside the algorithmic system). And this then means that AI is limited in what it can do in regard to consciousness or self-awareness such as humans possess. AI fans believe that someday AI can duplicate human intelligence – Gödel’s Incompleteness says “no way!” There will be more to say about this conflict!

In his final chapter on “Strange Loops, Or Tangled Hierarchies,” Douglas R. Hofstadter gives a grand windup, which may make you want to read this amazing, mind-boggling book again! Now, with my background of GEB, I look forward to reading Hofstadter’s more recent works on AI and strange loops.