A review of the book Godel, Escher and Bach

Godel, Escher, Bach; An eternal gold braid

Vintage books, 1979

 

A 1999 edition is available on Amazon.

I recently read “spacebookthenovel” by Robert Allan Richardson in which he referred glancingly to this astounding book that I had read many years ago.  I could not resist it, I searched my library and there is was among my mathematics books.  It is not an easy read by any means and as I looked at it I saw that one of our Labs had found it an equally tough item to chew on.  She had managed to make inroads on the lower left corner but thankfully left the rest intact.

First let’s look at the three individuals.  Opening with J.S. Bach, Hofstadter examines this brilliant musician’s famous Musical Offering to King Frederick the Great.  It is what we would now call a fugue and it’s theme is The Royal Theme.  It is an extraordinary musical piece being a six part fugue.  To give you an idea of how extraordinary this is you could compare it to playing sixty simultaneous blindfold games of chess and winning them all.  In presenting his copy of the piece to King Frederick the following was inscribed on the first page: Regis Iusfu Cantio Et Reliqua Canonica Arte Refolula.   “At the King’s command, the song and the remainder resolved with canonic art”.  Ricercar are the initials of the statement but also an old Italian word meaning “to seek” (in modern Italian “cercare.”) In these musical types a single theme is played against itself.  A very simple example would be to sing what we call rounds such as “row, row, row your boat, or Frere Jacques.  The voices are sung by voices coming in at every fourth beat but they can also be sung inverting the melody, reversing it, echoing it, going up or down an octave, or even changing the time.  The central point is that the ricercar is RECURSIVE producing what Hofstadter calls “strange loops”.

The four CD set “The Glenn Gould edition of The Well Tempered Clavichord” with 48 fugues and preludes demonstrate this type of music very well. He was particularly renowned as an interpreter of the keyboard music of Johann Sebastian Bach. His playing was distinguished by remarkable technical proficiency and capacity to articulate the polyphonic texture of Bach's music. You might be interested in exploring this musical piece.

Next is M.C. Escher.  If you have ever seen any of his work you will remember they feature impossible constructions, explorations of infinity, architecture, and tessellations. Take  his pencil drawing of two hands, each drawing the other.  Another is his “waterfall” which pictures a canal of water starting under a waterfall that turns a waterwheel, proceeds through four 90 degree turns and ends producing the waterfall from whence we started.  Again a RECURSIVE resulting in that strange loop.

If you are interested in pursuing Escher’s art I recommend “The Graphic Work of M.C. Escher” available on Amazon for, interestingly $95 or $.065.  I can never figure out Amazon’s pricing but there are other editions of Escher and you will enjoy looking at his truly fascinating view of the world.

And now to Kurt Godel, the key figure in this wonderful book.  In our first two artists we could see a conflict between the finite and the infinite resulting in paradox.  In the 20th Century a mathematical equivalent was discovered , with enormous repercussions.  A Strange Loop was found by this brilliant mathematician.  In its simplest form it can be likened to the liar’s paradox proposed by the Cretan Epimenides to this effect: All Cretans are liars.  Cogitate on this simple statement for a moment.  An even simpler statement would be “I am Lying” or “this statement is false.”  It is self-referential and a Strange Loop indeed!  In Godel’s mathematical world his strange loop statement, which I will not produce here because it is in German and even translated to English you will think it still in German,  means  he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers set theory with the axiom of choice, that: If the system is consistent, it cannot be complete and the consistency of the axioms cannot be proven within the system. That is  Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. In fact, there is an infinity of true statements it cannot prove. Whew!

Turning now to the book it is centered around Strange Loops, those recursive auto references; that generate twisted hierarchies, which, according to the author, can explain much about Intelligence, Consciousness, DNA, symbols and meaning. It  spans Zen and Zeno, Bach and Cage, Escher and Magritte, Gödel and Turing, Cantor and Russell. The auto-references allow him to jump up and down, and move in stimulating vortexes, culminating as a ricercar in 6 voices in a digression about free will, life and death at various levels

Also, each "serous" chapter is prefaced by an amusing dialogue between Achilles and a Tortoise (from Zeno's "paradoxes') and other droll characters. 

All in all a fascinating book. It is a golden braid intertwining looks at logic, philosophy, consciousness, music, art, biology, computers, and countless other topics through the prism of concepts of uncertainty, incompleteness, and self reference. 

If you decide to buy this book, and you should, get the 1999 edition.  Hofstadter’s forward does a much better job of preparing you for what follows than did his original.  And remember, you don’t have to read it all in one pass.  Do as my Lab did: chew on small portions and enjoy.