On Christmas, I posted Godel, Escher, Bach: Introduction suggesting that this book would be a good one to discuss in a series, and discussing the cover and table of contents of the book. Go read that diary first, please.
Today, I will diary about the chapter in the book entitled
Introduction: A Musico-logical Offering.
Later diaries:
second
third
fourth
fifth
First, let's look at the overview of that chapter (on p. viii in my edition)
I don't think it violates fair use to quote it (especially as I hope this series will get more people to read the book).
The book opens with the story of Bach's Musical Offering. Bach made an impromptu visit to King Frederick the Great of Prussia, and was requested to improvise on a theme presented by the King. His improvisations formed the basis of that great work. The Musical Offering and its story provide a theme upon which I 'improvise' throughout the book, thus making a sort of 'Metamusical offering'. Self-reference and the interplay between different levels in Bach are discussed; this leads to a discussion of parallel ideas in Escher's drawings and then Gödel's Theorem. A brief presentation of the history of logic and paradoxes is given as background for Gödel's Theorem. This leads to mechanical reasoning and computers, and the debate about whether Artificial Intelligence is possible. I close with an explanation of the origin of the book - particularly the why and wherefore of the Dialogues.
Already we begin to see how Hofstadter intends to tie together the three people in the title. Clearly, the idea of self-reference is key.
Now, to the chapter itself (pages 3 to 28 in my edition, which is the 20th anniversary one; I think the page numbering is the same in the original).
One problem with this book is that you can't hear Bach's music through its pages. The main piece referred to in this chapter. Here is what Wikipedia has to say about that piece, and, in the external links section are several renditions of all or part of it that are available online. One version is
You can, of course, duplicate art in a book, and Hofstadter provides many examples of Escher's art. Here is MC Escher's official website. It includes a picture gallery, with all of his work (I think it's all of his work, anyway). His early work shows little of the characteristic self-reference, the first work where I see this starting to happen is Procession in Crypt, and it is very strong in Hand with Reflecting Sphere (both of which are in the "Italian Period"). In his later work, there are many examples, many not reproduced in GEB. How about "Magic Mirror" (in the Back in Holland section)?
Kurt Gödel was a fascinating person. I wrote about him here. If you want to read a single book about him, I think the best is Rebecca Goldstein's Incompleteness: The proof and paradox of Kurt Godel. Also highly recommended is the old book by Nagel and Newman Godel's Proof.
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Take a look at Waterfall on p. 11. What's going on here? Water can't endlessly descend, and clearly the water at the top (just before the fall) is higher than at the bottom of the fall.... yet where does it go up?
I'm not sure, but I think where Escher hides the rise is in the bottom of the track that the water flows through. In the first section of the track, look at the pillars ... near the beginning of the track, the pillar is short. At the end, it's taller.
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The quote from Gödel on page 17 shows the author's playfulness (Hofstadter, that is, not Gödel. No one ever described Gödel as playful). "Perhaps you feel that it might as well be in German anyway" :-)
But even his more normal English may strike some as being somewhat opaque. A lot of the rest of the book is about what it means. But there's quite a bit in this chapter. Of Gödel, Escher and Bach, Gödel is the one I know most about, so if people have questions, I'll try to explicate.
One note is that Principia Mathematica is an incredibly dense book. I've never read it. Here is the Wikipedia page. Look at the quote from the book on that page. Whew! And note that it takes 379 pages of such writing to get to 1 + 1 = 2, but that 'the proof isn't completed until the second volume'!
On a personal note, Russell is an amazing guy .... radical to the core, and one of the few people to protest both World War I and Vietnam.
Near the end of this chapter, Hofstadter talks about the structure of the book, a counterpoint between the chapters themselves and the 'dialogues'. The first dialogue goes with Chapter 1, which will be next week. But I'll discuss it here, to whet your appetites.
"three part invention" is a play on a set of compositions by Bach (entitled three part inventions, but also known as sinfonias) which go along with the two part inventions....wikipedia linkincludes performances. They are contrapuntal pieces in two and three parts, respectively. Hofstadter's version includes the Tortoise, Achilles, and Zeno, and is structured around one of Zeno's paradoxes. But it's also full of self-reference and curious play .... Achilles and the Tortoise were creations of Zeno, yet they discuss Escher, who lived millenia later, and Zen, which is from the other side of the world. Incidentally, in his preface to the 20th anniversary edition, Hofstadter notes that he "finds Zen not only confusing and silly, but inimical to my core beliefs" ... but that he also finds it refreshing.