Gödel, Escher, Bach

I’m working my way, day by day, through the Pulitzer Prize-winning classic, "Gödel, Escher, Bach"; it’s one of the seminal books on the search for the "I" within us, and a prodigious work (pray for me).

By Dale Conour

I can’t resist books that promise a glimpse at the big picture, that connect the dots.

Some 20 or so years ago, a book came out that relatively few people took on and understood completely, but enough did for it to win the Pulitzer prize.

The book was called Gödel, Escher, Bach: An Eternal Golden Braid, and when I recently came upon a 20th-anniversary edition of it in Bell bookstore in Palo Alto, California, it was like unearthing some ancient lost talisman.  It was the second time I’d just chanced upon this book. Many years ago now, during one of my hours-long hauntings of a local bookstore, I’d discovered it. I hefted the heavy book, studied the covers and flaps, and just knew there were answers in there for questions I hadn’t even thought of asking yet. But I also innately understood it was way over my head. I wasn’t ready for it. I’m probably still not, but what the hell, right?

GEB is one man’s search for the origin of our consciousness, answering the question many are still searching for today, as if his book never existed. Where, within the brain, in the midst of all the electrical activity, neurons firing, synapses made, does the "I" come from?

As many seem to think, some mysterious, magical as yet undiscovered place deep within the brain? Writes author Douglas R. Hofstadter (who I’m in complete awe of):

"As I see it, the only way of overcoming this magical view of what "I" and consciousness are is to keep on reminding oneself, unpleasant though it may seem, that the "teetering bulb of dread and dream" that nestles safely inside one’s own cranium is a purely physical object made up of completely sterile and inanimate components, all of which obey exactly the same as those that govern all the rest of the universe, such as pieces of text, or CD-ROM’s, or computers. Only if one keeps on bashing up against this disturbing fact can one only slowly begin to develop a feel for the way out of the mystery of consciousness: that the key is not the stuff out of which brains are made, but the patterns that can come to exist inside the stuff of a brain."

Hoftstadter’s talking about self-referencing loops, crazy weird loops, for me the most effective metaphors being training a video cam on a mirror and the resultant endless images of the camera filming a camera filming a camera or the famed Escher lithograph of the hands drawing themselves.

Here’s the goal. I’m going to make my way through Gödel, Escher, Bach, and every day I’m going summarize what I glean from it or how I'm confused by it; I’m going to preserve my thoughts about it here. If anyone out there has read it, loved it or hated it, I’d love to hear from you.

Wish me luck: I’m going in.

Links: Gödel, Escher, Bach, from Books Inc

GEB II

My first foray into GEB has rewarded me already: I have a much deeper respect for "Row, Row, Row Your Boat."

Hofstadter begins the book by relating how J.S. Bach came to write a now-mythic set of canons for Frederick the Great, King of Prussia, known as the "Musical Offering." In so doing, he reminded me (who must now embarrassingly admit that I started out college as a music major), what a canon is.

A canon is a theme played against itself. You start to sing "Row, Row" and then after a few beats, a new voice begins the tune. The trick in writing even so simple a tune as Row or "Three Blind Mice," of course, is that it has to sound good when played against itself.

And from there, there are ever more complex variations—voices performing the theme in higher notes, at faster or slower speeds. The theme must sound good alone, or in harmony with itself, and more.

I am reading this, and thinking, well, okay, I’m learning about Frederick (and imagining what the world would be like today if more leaders combined the modern belief that prisoners shouldn’t be gutted in public squares with the old school love of the arts) and Bach, and the secret genius behind "Row, row, row your boat," and that’s all good, and I’m sure the context for this is coming, when Hofstadter points out:

"Thus, each note in a canon has more than one musical meaning: the listener’s ear and brain automatically figure out the appropriate meaning, by referring to context."

And suddenly a small door opens in my brain. A little fresh air blows through. Something’s coming.

"Notice that every type of "copy" preserves all the information in the original theme, in the sense that the theme is fully recoverable from any of the copies. Such an information-preserving transformation is often called an isomorphism, and we will have much traffic with isomorphisms in this book."

To end this first dip into Bach, Hofstadter refers to one canon in Musical Offering, "Canon per Tonos" as the "Endlessly Rising Canon." Bach managed to compose a piece in C minor that not only seamlessly ends a key higher in D minor, but allows the performer to then repeat the canon in D minor and move up to the key of E and so on, ad infinitum.

"In this canon, Bach has given us our first example of the notion of Strange Loops. The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started.."

Next: A first look at Escher

GEB III

The Strange Loops return in the second section of Hofstadter’s "Introduction: A Musico-Logical Offering," only this time it’s an artistic representation of them rather than a musical one. And our champion now is the great graphic artist M.C. Escher (1902-1972)—the guy who drew all those mind-blowing, physics-defying, mathematics-loving illustrations, like "Waterfall" shown here.

You look at it and think, "Hmmm—waterfall cascading to waterwheel within a building, and circling around again, okay, cool I guess." But then you look closer and realize the water is not really rising up as it courses along the channel..but it is.

But it isn’t. At least it ends up at the top again. Somehow. Whoa.

Waterfall, Hofstadter points out, is a six-step endlessly falling loop just like Bach’s "Canon per Tonos" that I wrote about last time. "The similarity of vision is remarkable. Bach and Escher are playing one single theme in two different "keys": music and art." The author goes on to talk about Escher’s various treatments of Strange Loops in his works, and how he varies from tighter loops to looser ones.

More interesting at this point is his explanation that, "Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in an finite way?" And infinity plays a large part in Escher’s work:

"In some of his drawings, one single theme can appear on different levels of reality. For instance, one level in a drawing might clearly be recognizable as representing fantasy or imagination; another level would be recognizable as reality. The two levels might be the only explicitly portrayed levels. But the mere presence of these two levels invites the viewer to look upon himself as part of yet another level; and by taking that step, the viewer cannot help getting caught up in Escher’s implied chain of levels, in which, for any one level, there is always another level above it of greater "reality", and likewise, there is always a level below, "more imaginary" than it is. This can be mind-boggling in itself. However, what happens if the chain of levels is not linear, but forms a loop? What is real, then, and what is fantasy? The genius of Escher was that he could not only concoct, but actually portray, dozens of half-real, half-mythical worlds, worlds filled with Strange Loops, which he seems to be inviting his viewers to enter."

And to think we’re only getting started.

Next: Our first look at Gödel.

GEB IV

And now comes the math. Math and I have had a very unsatisfying relationship for a very long time. I appreciate math,  and I respect it, I really, really do. But math, I’m just not into you.

My problem with math has historically fallen into two general areas. One, a lot of mathematics has always seemed to exist in some self-referential world that has nothing to do with the world I live in. Two, I don’t like questions with only one right answer. No "perfect" system for me, thanks, I’ll take some relativity.

So as I struggle through this next section of Hofstadter’s introduction covering the great German mathematician Kurt Gödel and his role in the author’s theory of consciousness, it’s a bit ironic to learn that Gödel’s contribution was to endow mathematics with the power of self-reference, introducing what became known as Gödel’s Incompleteness Theorem.

The Theorem proved that statements of number theory can be true even if they can’t be proven within a given "fixed system of number-theoretical reasoning," and did so by implanting our new best friend the "Strange Loop" within such a system.

The actual Theorem, unless you are a math god, will not make sense. (And if you are a math god, help me with all this, and stop sitting there chuckling derisively.) Even the Theorem as paraphrased in English by Hofstadter takes awhile to get. But basically what Gödel did was convert a self-referential statement in language, the Epimenides paradox

"This statement is false"

...into a self-referential mathematical statement (no one had even considered doing this, let alone attempted it) that created a paradox within a system (Principia Mathematica) constructed some 30 years before precisely to vanquish paradoxes, and buttress the foundation of logic and its Siamese twin, mathematics.

Gödel cracked open the foundation to expose a truer one, and rocked the world. He helped usher in a new era in mathematics (as well as logic and philosophy) that actually helped us further understand the real world and how it works, ultimately leading to concepts like quantum mechanics and...relativity.

Okay, Math, maybe I’ve been wrong about you. And since you and I are going to be together for the next 743 pages, I guess we'll just have to learn how to get along.

Next: Babbage, Computers, Artificial intelligence, the "Golden Braid"...and then we really get going

GEB V

How human can a machine ever be? How machine-like are we? Author Douglas Hofstadter winds up GEB’s "Introduction: A Musico-Logical Offering" with a few thoughts about Artificial Intelligence, and its intrinsic paradox:

"Computers by their very nature are the most inflexible, desireless, rule-following of beasts. Fast thought 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 to urge each reader 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 the formal and the informal, the animate and the inanimate, the flexible and the inflexible."

Indeed, Hofstadter notes that even so forward a thinker as Lady Ada Lovelace (daughter of Lord Byron), who predicted back in the 19th century that Charles Babbage’s "Analytical Engine"  (shown here) if/when built and running (which never happened), could potentially produce complex music compositions and graphics, couldn’t imagine it originating anything, only performing as ordered. (The woman, by the way, sounds fascinating; follow the link below to learn more about her.)

Referring again to the genius of J.S. Bach, Hofstadter quotes from theologian Johann Michael Schmidt, who, in  1754, four years after the death of the composer, made his case against those equating men to machines (and vice versa):

"No one has yet invented an image that thinks, or wills, or composes, or even does anything at all similar. Let anyone who wishes to be convinced look carefully at the last fugal work of...Bach...and let him observe the art that is contained therein; or what must strike him as even more wonderful, the Chorale which he dictated in his blindness to the pen of another: Wenn wir in höchsten Nöthen seyn...Everything that the champions of Materialism put forward must fall to the ground in view of this single example."

As Hofstadter points out, it’s been 200 years and this particular battle still goes on: "I hope in this book to give some perspective on the battle." He does this by weaving an "Eternal Golden Braid" out these three strands we’ve discussed so far: Gödel, Escher, Bach.

Next: Tortoise, Achilles, and MU.

Links: More on Ada Byron, Lady Lovelace

GEB VI

Curiouser and curiouser becomes the learning as I drop further down the GEB rabbit hole. Hofstadter begins what will be the pattern of the entire book: ping-ponging between "Dialogues," featuring the characters of Achilles and the Tortoise and delving into his concepts through metaphor, and "Chapters," addressing the same concepts through more formal means.

Achilles and the Tortoise were first employed by ancient philosopher Zeno of Elea (he predated Socrates and Plato), who used them to illustrate his famed paradoxes, and then resurrected millennia later by Lewis Carroll to show off a paradox of his own (to come later in the book, promises Hofstadter).

Hofstadter retells Zeno’s account of a race between Achilles and the Tortoise in which the philosopher puts forth the following argument: Given a head start and constant, infinite motion, the Tortoise will always stay ahead of the fleeter Achilles if he merely vacates each point in space before Achilles arrives there. After all, Achilles must reach the point where the Tortoise just was before he can then advance to the one where the Tortoise is now.

It’s a variation of Zeno’s "dichotomy paradox" that supports his theorem that Motion is Inherently Impossible (Motion Unexists) because, in traveling from A to B, you have to go halfway first, but first halfway of that, and halfway of that, and so on, to infinity.

Since Zeno managed to get around despite his clever argument, he was really using these paradoxes to show that space and time clearly aren’t segmental, that they are continuous, and therefore, the linear concept of Motion must be a mental, not a physical construct.

Hofstadter’s variation of the race includes A and T remarking on a Zen koan in which two monks argue about a flag. "The flag is moving," says one. "The wind is moving," counters the other. A master happens by, and says they’re both wrong: "Mind is moving."

Is it possible I’m oversimplifying all this? Uh, yeah...

And of course, the author’s version of the tale also has loops. The two characters discuss Zeno’s paradox only to have Zeno show up and encourage them to have the race that they were just conversing about to "empirically" prove the theorem that he’s already used them metaphorically to prove.

And they will be racing toward a distant flag that resembles Escher’s Möbius Strip I—something they both acknowledge can’t really be there because it hasn’t come into existence yet (from their perspective).

What’s it all mean? We’ve covered plenty to chew on already. But the Chapter that follows lays out the underpinnings of a "formal system." Formal systems house theorems (like Zeno’s presumably). Do I understand how the first Dialogue and Chapter completely relate? Not completely yet. But hopefully I’ll be much closer by the next time I report on this...

Hang in there with me: We’re not only chasing the "I," we're attempting to understand a new version of "reality." Nothing like a good brain stretch.

Next time: The puzzle of MU, and the importance of jumping out of the system

Comments

[hadi]

I can't wait for the continuation.
I am having a hard time to read EGB.
Hoping reading your blog will help me.

Thanks for this.