Godel, Escher, Bach - Intro - Ch. 4

Hey all, I'm just posting a thread to get us started, and also to have an actual thread to point people to who want to get in on the discussion. So... discuss away! ;)

Hi!

For a start, shall we set Sunday, April 6th, as the date for finishing the Introduction, Three Part Invention and Chapter 1? It's not that many pages (3-41) and provides a taste of the book's 'flavor'.

For my part, I'll try to add comments on the musical references and also on the word play. (Sample: A great, listenable canon is the strict 2-voice 'Witch's Minuet' movement of Haydn's d minor string quartet ('Fifths'), Op. 76, No. 2, 3rd movement.)

Are introductions in order? I'm Jim. Location: Staten Island, NY. I'm an old retired geezer. An introduction to my interests might well be the little piece of fluff I wrote exploring self-referent sentences through a self-referent poem. It's on my profile page.

Thanks for the message, Jim.

Hello all -

I'm Sarah, from the UK. Still earning a living cutting code, but hoping to retire, or at least cut back, in a few years. Pic of me on my page.

I don't have my copy yet, so I shall be lagging somewhat behind. In advance of that, is there a list of music I should look out for? I don't have much Haydn, tho' quite a lot of Bach. (I'm a sucker for Das Wohltempierte Klavier and the violin partitas.)

Great idea! And Sarah - how do you get the cat & the bird to get along so well??

I'm Dani, and I'm working on my doctorate in philosophy in Houston, Texas. (Yes, Texas really is like "another country." Needless to say, I'm not a native...) I love logic, but confess to realizing the first time I started this book that my knowledge of musical theory is abysmal.

I had posted this in the other thread, when we were still kicking around the idea of a group read, but maybe it's more appropriate here. The Lewis Carroll article, What the Tortoise Said to Achilles is the inspiration for the dialogues in between the chapters of GEB, and you can find it online here.

The cat's translation of 'get along so well' is "What bird?", and bird's translation is "What cat?" (though she does eye Tilly out of the corner of her eye). Actually, its a case of MAD (you remember Mutual Assured Destruction, I'm sure) - four paws with claws equates to one beak like a ratchet.

Thanks for the link to Lewis Carroll. I hadn't read that before and now I know why Gerald Durrell's tortoise was called Achilles in My Family and Other Animals.

For those not familiar with fugues, I've taken one apart. It's posted on my profile page. Sorry 'bout the formatting.

Hello!

I'm a grad student working on ancient Greek mathematics, and like several people here, I have a very limited knowledge of music.

I considered saying "My name is....", but decided it was pretty obvious :)

I've started reading GEB before and I'm hoping that this discussion will be enough of a refresher that I won't need to start over again, so I may not be able to contribute very much to the discussion of the earlier chapters.

Hi, I am an electrical engineer working in the motor manufacturing industry, I am doing a lot of research and advanced design work. I started GEB as a high school student, didn't understand a lick of it, so I gave up. Hopefully I am mature enough now to understand it.

I'm Jim, currently a system engineer, formerly a physicist. I'm also a musician, so that part of the work is fascinating to me. It's been a while since I read GEB last, so I'm looking forward to dipping in again as a group read!

I'm a computer programmer, currently unemployed. I read GEB in college or soon thereafter, some 20+ years ago and found it extremely interesting although I know I skimmed some of the more technical areas. It helps that one of the math courses I took in College was on Goedel's theorem.

Has anyone been able to generate MU from MI yet?

My copy of Godel, Escher, Bach was given to me by my mother in 1980. I think it is Douglas R. Hofstadter's best book.

I have sung a little bit of Bach and it can be difficult, but there is a great deal of satisfaction if your choral group gets it all right. This sometimes may only happen once, and that frequently is at the main performance.

Many singers get it by counting beats, but I find that difficult. I try to sing in relation to what the others are singing. I started out singing tenor, but had no support, and then dropped down to bass-baritone, where I did have support, and have done much better.

I'm David. I teach (English language, writing, literature) at a university in Japan, and write about books for a few different venues, most regularly the Japan Times.

My interest in music and in philosophy is strictly an amateur's (though I'm passionate about the former). I'm looking forward to taking down GEB from the shelf where it's sat for twenty years or so and learning from those of you more knowledgeable than I about music and philosophy.

What a wonderful mixed bag of readers!

For those coming to GEB for the first time, I've started a series of sidebars on my profile page.

For those interested, I got a copy of the book J. S. Bach's Musical offering: History, Interpretation, and Analysis a few years ago as a supplemental text. I also tried to figure out counterpoint but hadn't gotten around to it, I am ashamed to say.

So there is another book to explain the specific musical reference in GEB.

Basic counterpoint is done with four voices, each singing its own melody. After the baroque period much of four-part singing was with one voice, usually the soprano, carrying the meolody, and the other three voices providing the harmony.

To make counterpoint more difficult, each voice may sing the same text, but sometimes they may not all start together. So a line of text like, "And He shall reign forever and ever," may start off with only the sopranos singing, and the altos may come in with "and" while the sopranos are singing "forever". Or the sopranos and tenors can start together, and then the altos and basses come in later. And then you can do another permutation of this.

Everyone once in a while you may devolve into 8-part harmony (high soprano, soprano, mezzo-soprano, contralto, tenor 1, tenor 2, baritone, base). This may occur when the text seems to want to have a call and an answer, or when you might want some bit of sung commentary. But if you're listening to what's being sung in Latin, sometimes German, you have really no idea of what's going on. And frequently the singers don't either, they are concentrating on counting their beat, so as to come in right. However, it is usually arias that guess into all of this, and since every line of text gets repeated quite a bit, you are likely to hears some short stretches of only voice singing, so if you can process German or process Latin, you can get it. The reason people like Handel's Messiah so much is that it is one of the very few oratorios written in English, but Handel is not our subject.

My name's Robo. I did a philosophy degree, and work in IT. I've read GEB twice, the second time in a book group. That group had a few of us that were somewhat educated in the philosophy and maths aspects of the book, but nobody that knew much about Bach. So I'm delighted to see that there are some people here who can give more insight into that side of things.

Robo:

Cut a deal with you? I'll discuss the WTC fugues if you'll define existentialism! ;-))

Hello all-- my name is John. I am an IT and business systems analyst with degrees in philosophy and psychology and an incomplete MS in Logic and Computation.

GEB was a major influence in my early undergraduate education, but I have not reread a word of it until now. In the intervening years I have studied many relevant topics both in and out of college (recursion theory, history of analytic philosophy, epistemology, philosophy of mind, AI, etc.). I look forward to revisiting GEB with a new (possibly more critical) perspective and the good company of this diverse group.

Best Regards.

>10 jjwilson61: I thought I had one. I've found my mistake, though - an extra I got dropped inadvertently in the move from line 11 to 12. Now I feel very silly. I've edited this post extensively after finding my mistake.

Here's a more formal presentation of the derivation rules, if it will help anyone:

Where Φ and Ψ represent any partial strings;

I:
ΦI ├ ΦIU

II:
MΦ ├ MΦΦ

III:
ΦIIIΨ ├ ΦUΨ; or
ΦIII ├ ΦU; or
IIIΦ ├ UΦ

IV:
ΦUUΨ ├ ΦΨ; or
ΦUU ├ Φ; or
UUΦ ├ Φ

Note that only rules III and IV are replacement rules which can be used at any location within a theorem.

Also, the Lewis Carroll article I linked above actually shows up in the text, as the dialogue between Chapters 1 & 2.

I'm not sure there's a whole lot for discussion in the intro & first chapter (though I'm sure to be proven wrong in that!), it's really just a setting of the scene and introduction to the terminology that will be used throughout the text. The most helpful bits for me are the explanations of canons and fugues at the very beginning of the intro. The musical scholars amongst us probably didn't need it much, though I guess a refresher course can always be good :)

Note that GEB is very carefully written. The choice of MU for the puzzle was not at all random, but specific for its connotations, one of which provides an insight. You may find the following link of interest:

http://www.ibiblio.org/zen/gateless-gate/1.html

So , does the discussion start tomorrow?

Sure does. Shall we start with the Intro, the Three Part Invention and Chapter I? How are things coming along with the MIU system? Remember, this is an exercise in working in a formal system and also viewing it from 'outside'. (The Two Part Invention and Chapter II will expand on this concept.) Sometimes, viewing a problem from a different perspective, or from 'outside', can lead to what I like to call an 'Aha!' experience.

Because I wanted to finish Roberto Bolano's delightful The Savage Detectives before diving into GEB I'm a bit late out of the gate with comments.

* First I applaud Hofstadter for his lucid and witty prose. When he writes, for example, of Godel's Theorem:

"What the Theorem states and how it is proved are two different things. . . . The Theorem can be likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem,"

I can only marvel (and chuckle) at the aptness of the metaphor and how illuminating it is.

* "Provability is a weaker notion than truth" Hofstadter writes on page 19 (I'm using the old, gold-covered, paper-backed Vintage edition).

Yes!

Perhaps this is not unrelated to the certainty one has that (one's favorite artist) is a greater painter than (an artist one consider a total hack), but that, however powerful an argument one can build in support of that position, one cannot prove it.

* On pages 19-20 Hofstadter writes: "How could there be many different kinds of 'points' and 'lines' in one single reality? Today, the solution to the dilemma may be apparent, even to some non-mathematicians." I am one non-mathematician to whom the solution is not apparent. Can anyone enlighten me?

* Is Hoftadter having us on when he refers to a scholar named "Kneebone" on page 19? He tells us in the list of references that Kneebone's book is "solid."

* I found Hofstadter's introduction to logic--probably boring those out there better educated in philosophy than I--lucid and helpful. I look forward to seeing how he will complicate things as we go along.

And I look forward to seeing what the rest of you have to say about this week's read.

What's our next "assignment"?

>24 dcozy::

Mathematical Logic and the Foundations of Mathematics: An Introductory Survey, G.T. Kneebone. London, New York. D. Van Nostrand, 1963.

'Points' and 'lines' refer to (at least) those in the three modern geometries: traditional Euclidian and the non-Euclidean geometries' of Riemann and Lobachevsky. In the non-Euclids, lines aren't necessarily 'straight' and 'parallel' lines can pass through a point.

Nb: Do make sure that you carry away from Chapter I a solid concept of a formal system and what it means to look at a formal system from 'outside' or 'above' it. Later on, keeping these two things straight will become a tad difficult.

I'm of the opinion that we don't need specific dates by which people are "supposed" to have read a section by. We all agreed that roughly a chapter a week would be a good pace, but I'm very much opposed to setting specific dates and saying people have an "assignment" to read certain sections of the text by then, or saying that they can't read faster than that and begin discussing when it suits them for that matter... maybe we could try and keep this just a little more relaxed and a little less formal?

Just my thoughts ;)

Comments on Intro, the Three Part Invention and Chapter I:

• To my delight, Hofstadter is as playful a writer as I remember from my early undergrad days.


• * His brief history of mathematical logic is a bit unfair to Russell's theory of types. Russel's paradox is a syntactic paradox that results in contradiction, which is lethal to classical logic, as it allows any theorem to follow, thus negating the "truth-preserving" nature of logic. His simple type theory led to a ramified theory of types with interesting interrelations with category theory and set theory -- see Type Theory for a more complete picture of type theory and its descendants.


• * A small frustration from the professional community, related to much of GEB and his later work, such as I am a Strange Loop: Hofstadter has spent years on the idea that self-reference, specifically in the form of "strange loops" are at the core of identity and intelligence, yet he has not made much headway in characterizing this in any way that can be utilized by logicians, mathematicians, computer scientists, or cognitive scientists. Meanwhile, different approaches have advanced, such as paraconsistent logics without comment from DRH.


• * I had largely forgotten about the MUI-system. What a clever exorcise!


• * At the end of Chapter 1, Hofstadter reviews decision procedures (a.k.a. algorithms). and proposes a simple theoremhood test. This is the primitive basis for what is called automated theorem proving, which has become quit a useful field for mathematicians and computer scientists. See, An Overview of Automated Theorem Proving for a clear summary of ATP and its use.


• * For those interested in learning more about Zen Buddhism, Zen Flesh, Zen Bones is a good, often humorous introduction, or you can get the bul of the text online: 101 Zen Stories and The Gateless Gate.

Torus34 suggests:

"Do make sure that you carry away from Chapter I a solid concept of a formal system and what it means to look at a formal system from 'outside' or 'above' it. Later on, keeping these two things straight will become a tad difficult."

Yes. I got that this was an important concept.

Hofstadter writes: " . . . there are case where only a rare individual will have the vision to perceive a system which governs many peoples' lives, a system which had never before even been recognized as a system; then such people often devote their lives to convincing other people that the system really is there, and that it ought to be exited from!"

It strikes me that this ability to perceive systems that others don't, and to formulate ways to exit those systems is a hallmark of genius: Einstein seeing outside the box of Newtonian physics, Joyce seeing outside the box of the Victorian / Edwardian novel, Darwin seeing outside the box of pre-Darwinian biology, Beethoven seeing outside the box of classical modes of composition.

And it's certainly fine with me if we all read at our own paces. I've just begun chapter 3.

>28 dcozy:

It's interesting to me that you alit on that passage about stepping out of systems; it stuck out to me too.

Part of it is that my brain tends to function most comfortably in the ethical/political mode, and the passage seemed relevant to that; but the other thing that struck me about it was the Kuhn-ian undertones of the statement. Your thoughts about Einstein & Darwin seem to sync up pretty well to my intuition there - and I think the analogy is easily extended to the realms of the artistic as well.

I think the ability to step out of system and observe is a unique talent but I am not so convinced that it is a hallmark of genius. It certainly does help when you get the big picture through observing. Other cliches used to describe this ability are: seeing the forest and not just the trees, thinking outside the box, working at 10,000 ft, etc. And I do believe Godel's theorem has been appropriated or more often misappropriated as the governing principle for this line of thought. Godel, being a mathematician, is spare and precise with the language, whereas more people who extrapolate can be verbose and imprecise, thereby introducing extraneous language and details not strictly intended in his theorem.

I'm sure pw0327, in message 30, is correct about Gödel and his theorem, but my comment, that the ability to see outside the box (my preferred cliche) might be a hallmark of genius, was a response to a remark of Hofstadter's, not Gödel's. My notion did not, as pw 0327 correctly notes, derive from any deep understanding of Gödel's theorem. I don't possess such an understanding; I have, however, considered the lives of a few geniuses, and it does seem to me that such geniuses are often able to step back and take a larger view than their contemporaries.

Of course to become a genius it is not sufficient to think outside the box. One must also have the drive and ability to put one's insights to fruitful use.

And I suppose there are also people who work entirely within the systems available to them in their times, but do whatever it is they do with so much more skill than their contemporaries that we are happy to call them geniuses, too.

> 31

That is my point exactly.

Genius ==> able to step out of system.
Ability to step out of system =|=> Genius

It can be useful, in reading a dense and difficult (also lucid and fascinating) book such as GEB to isolate what the author takes to be his or her purposes and goals in producing such a tome. Here are a few of what appear to be Hofstadter's that I've noted in the first five chapters:

"Here one runs up against a seeming paradox. Computers by their very nature are the most inflexible, desireless, rule-following of the beasts. Fast though they may be, they are nonetheless the epitome of unconsciousness. How, then, can intelligent behavior be programmed? Isn't this the most blatant of contradictions in terms? One of the major theses of this book is that it is not a contradiction at all. One of the major purposes of this book to urge readers to confront the apparent contradiction head on, to savor it, to turn it over, to take it apart, to wallow in it, so that in the end the reader might emerge with new insights into the seemingly unbreachable gulf between formal and informal, the animate and the inanimate, the flexible and the inflexible" (p. 26).

"Do words and thoughts follow formal rules, or do they not? That problem is the problem of the book" (p.46).

"A very important question will be . . . whether it is theoretically possible to attain the level of our thinking abilities by using some formal system" (p.60).

"In my opinion, in fact, the key elements in answering the question 'What is consciousness?' will be the unraveling of the nature of the 'isomorphism' which underlies meaning" (p.82).

"We keep on running up against 'sameness-in-differentness', and the question

When are two things the same?

It will recur over and over again in this book. We shall come at it from all sorts of angles, and in the end, we shall see how deeply this simple question is connected with the nature of intelligence" (p.148).

Perhaps reminding ourselves of Hofstadter's claims, and of the questions he thinks are important, will help us to keep our eyes—our minds—on the ball.

Re: Genius and thinking outside the box.

I recall once running across a definition of genius: the ability to see first that which is obvious to everyone afterward. Unfortunately, I don't recall the author.

In music, we call Wagner a genius for thinking outside the box of his (musical) time. Same with Haydn, generally credited with formalizing the string quartet. Mozart, however, used only those forms which were currently available and did little to stretch their limits. In terms of form, at least, Mozart did not think outside the box.

Sidebar: Someone once noted that if we ever receive signals from outer space, we should reply by sending back the music of Bach, Beethoven and Mozart. Bach, to show our intelligence. Beethoven, to demonstrate that we possess emotions. And Mozart to prove beyond any doubt that we are civilized.

#34

Was John Cage a genius for thinking outside the music "box"?

Is Brahms a genius for writing a requiem, not using the traditional texts of the Latin liturgy, but using his own selected texts in German? As I write this, I'm thinking maybe he was simply following Schubert, whose Deutsche Messe used words that are a considerable variant on the traditional mass.

35>

Is it possible that the criteria for 'genius', even when stated in very broad terms, is different for different disciplines? And thank you very much for de-rutting my thinking on this!

OK, I've sort of been holding my tongue about Hofstadter's slightly condescending attitude, whereby he thinks mathematics is somehow better than, and prior to, philosophy.

But I find his requisitioning of the term "recursive" to describe certain figures to be a just plain bad idea. Clearly, he's also working with the more traditional meaning of the word, as it applies to proofs and system definitions. But this double-use of the term is needlessly confusing. (And I get that its intentional because he thinks that its neat to have recursively enumerable sets which aren't recursive. Yay for loops. It seems to me to just muddy the water, and make things confusing which don't need to be.)

OK, I'm feeling a little feisty today. ;)

I can't find the pattern in the set on p. 73. The best I can come up with is that the differences between the differences between the numbers seem to have the pattern of 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2.... but I feel like there's got to be something simpler there which I'm flat overlooking. Anyone got something better?

You know, I was reading chapters 2 and 3 yesterday (ok, I'm a little behind schedule...) and it struck me that he seemed a bit more smug about being clever than I remember from the last time I read GEB twenty-ish years ago. Maybe I was also a bit feisty yesterday! 8^}

I'm not certain I completely get how he's using the term "recursive". I understand the traditional meaning. Anybody got a reasonable definition of recursive as applied to art or music? Or are we supposed to get it from examples and context?

What I get for the way he's using it for art & music is that a piece of art, for example, is recursive if not only the foreground is a distinct picture/object, but the background also forms a distinct picture or object. Similarly for music, if you can find a melody in the harmony (I think?) or can make something out of the "empty spaces" between the notes.

His actual explicit definition is on p. 67:
A recursive figure is one whose ground can be seen as a figure in its own right.

(And I agree. I think "smug" is maybe a better word than "condescending" for the vibe I'm getting off him.)

edited to close a tag.

Hofstadter is a computer scientist and not an art historian or a musical theorist so the art and music parts should be a little closer to the surface than the mathematics part, so I had retty much lowered my expectations in that regard, even though I am not any of the three. His reference to recursion is definitely more along the lines of math/computer science. The Escher and Bach part of the exposition seem to me to be more of a: "check it out, this is kind of cool" statement.

I think as an amateur philosopher pontificating, he did well but I am sure philosophers won't think so, especially about his rigor and reasoning.

Smug is a pretty good word to use.

I didn't get the smugness. To me the tone conveys more excitement about what it is he wants to impart. If *I* were in a feisty mood I might say his prose calls to mind the sort of guy, eyes a little too bright, who collars you at a conference or in a bar to tell you about this amazing and important work he has done that no one else appreciates. Sometimes such spiels can be quite fascinating, and I feel, most of the time, that such is the case with Hofstadter.

I've now finished the first half of the book, and while I find each chapter interesting and (with little philosophy and less mathematics) I learn something from each, I'm not quite sure how he's going to tie together all the knowledge he has presented. I look forward to finding out.

Like philosojerk (#37) I am unable to "characterize the . . . set of integers (or its negative space)" presented on page 73 in a fashion at all elegant. I hope someone can help with this.

As illustrations of Hofstadter's points, Escher's pieces are fun to look at. Seeing them, though, reminds me how, once, having wandered into a show of Escher's work, moving from one such piece to the next to the next, I quickly became bored . Cleverness only goes so far in art.

And speaking of art I had the good fortune to hear, in Tokyo, Angela Hewitt play the first half of The Well Tempered Clavier last night. Though her Bach was, at times, too Beethovian for me, it was still a marvelous evening. I look forward to the second half tomorrow afternoon.

I'm at home for about 15 minutes between social engagements, but I went ahead and started a new thread, for chapters 5-9, right here. The general pace we set was roughly a chapter a week, but I get the definite impression that some people are ahead of that, and I know (for me, at least) there is often a strong inclination to jot down thoughts as you go, rather than wait for others to catch up. This way, if you feel so inclined, you can stick stuff there... and you only have to "wait" for the responses of others ;)

At least this way, people are less likely to forget what they had to say.

As I took the early suggestion that we should read at our own paces to heart and want to ask a question pertaining mostly to chapters 10 and 11 I've taken the liberty of starting a thread dealing with chapters 10-13. I don't know how to paste in a hot-link to it, but I'm sure you'll be able to find it.

Here is the hot link to Ch 10-13:
http://www.librarything.com/talktopic.php?topic=35331

To make a hot link:
You copy the address line while your are at that place.
You come back here.
Go to a separate line and type two spaces.
Paste the address line in, and type two more spaces.

Some one just sent me a Powerpoint of Octavio Ocampo art. Amazing stuff. A spectacular example of the Foreground/background effect.

http://www.davidairey.com/octavio-ocampo-and-art-that-makes-you-look-twice/

Here is but one web site with his work.