Godel Escher Bach: An Eternal Golden Braid

I’ve gotta admit, I have a real bias towards Douglas Hofstadter, one of the real Renaissance men of science and literature. He’s truly one of the rare cutting-edge scientists who can write a damn entertaining book. Despite being over 700 pages and touching on tons of topics, the central thesis of his most famous book isn’t quite as revolutionary as you might think, and is certainly nowhere near as daring the other scientists featured in my “I’ve got a theory” report. And this is a good thing — If I were a betting man (and I am), I’d put my money down on Hofstadter as the one most likely to be right.

“GEB” (as the folks-in-the-know call it) might be completely unique among popular-science books — I can’t think of another book that combines such deep scientific and philosophical issues with such a flair for the literary. This isn’t just a well-written science book (like Carl Sagan’s books), it’s a book with actual artistic and literary quality to it — if you squeezed out the sections that focus on science, you’d still have enough left over to cobble together a book for a college english class. The danger with a book like this is that there’s hardly anyone qualified to be a “target audience” — the uber-tech-nerds who truly understand all of the science and math likely don’t care about the literary styling (if they noticed it at all), and the poets who can appreciate Hofstadter’s penchant for a cool turn of phrase aren’t likely to get very far into the book without feigning a stroke so they don’t have to read about number theory. And yet, it still works — the book is good enough that nearly everyone recognizes it as a tour de force, even if hardly anyone understands the whole thing. Although perusing the reviews on Amazon confirms my suspicion — the very few people who didn’t like the book are either 1) crusty hard-core experts in mathematical logic or cognitive science who had no patience for the artsy stuff, or 2) artsy people who aren’t afraid to admit they didn’t get any of the science and so the book is lost on them.

Speaking of not getting it, it’s fun to see where this book winds up getting shelved in bookstores. Math? Good try, but the main point of the book isn’t concerned with math. Physics? Not even close. “General science”? Come on, you’re not even trying. Art? Music? Puh-leez. This is actually a cognitive science book, a sweeping statement about Hofstadter’s theory of the mind (and specifically of how consciousness arises), despite the extensive time spent on Bach (the composer), M.C. Escher (the artist), and Kurt Godel (the logician and mathematician), plus forays into solid-state physics, zen buddhism, computer programming, language translation, and some funky stuff about a video camera filming its own video screen.

The far-ranging topics of GEB reflects the far-ranging interests of Hofstadter. Get this — he’s a professor of cognitive science at Indiana University, with a joint appointment in computer science (impressive, but not that uncommon), but is also an adjoint professor in History and Philosophy of Science (makes sense), Philosophy (now that’s reach!), Psychology (eh?), and Comparative Literature (really?). Is there anyone else on earth who is listed on brochures from both computer science and comparative lit departments at their college? Oh, and his PhD is in solid state physics. And his first love in college was number theory. Now explain to me again, why exactly did you drop down to 12 credits that one semester in college, so you could “find yourself”? That cost you adjunct positions in at least two or three departments…

I’m being a little glib calling his work a “Grand Theory of Everything”, because while his theories cover a broad range including cognitive science, philosophy, math, and even humanities, he puts forth perhaps the most modest claims among any of the Grand Theorizers. While other folks I review (Roger Penrose, Stephen Wolfram, etc etc) put forth preposterous theories-of-everything that are generally ignored or ridiculed by the scientific mainstream, Dr. Hofstadter presents a very similar broad survey of the same kinds of topics (Godel’s theorem, human consciousness, theories of computation, etc.) but comes just short of actually connecting them together with some preposterous theory. His books aim to explain human consciousness, but he readily admits the topic is extremely complicated — in fact the real power of GEB is how clearly it depicts the intricacy and strangeness of consciousness. Nobody poses the problem of consciousness better than Hofstadter.

Hofstadter’s key tool is the analogy — telling one story to help tell another, seemingly unrelated story. The book presents a slew of analogies, examples of complex systems with many interacting and recursive parts that share some interesting feature with human consciousness. He walks through at least a half-dozen of these examples (including number theory, DNA, musical fugues, and even ant colonies) in detail. You get an in-depth tour of how DNA is used to make proteins, for example, ultimately to serve as an analogy to help you appreciate his view of how the mind works. Many of the examples feature simple interacting parts that collectively produce intricate, “emergent” behavior that you might not expect from such simple components. Most of them also involve what Hofstadter calls “strange loops”, meaning that some sort of hierarchy is present in the system, and goings-on in one level can “jump out” to affect other levels, often in mind-bending ways — like a Rubik’s cube solving itself, or an author becoming a character in his own book. Or a pack of gooey neurons becoming aware of itself. He brings up examples of systems from other domains to get you acquainted with some of the key ideas, just in time to broach consciousness itself by the last few chapters of the book.

One of these analogy-for-consciousness examples, Godel’s incompleteness theorem, is so compelling and rich in its own right that you can be forgiven if you come away from this book believing that it’s a book *about* Godel’s incompleteness theorem. It’s a theorem from number theory, discovered around the same giddy early 20th-century period that produced Quantum mechanics in physics and Cubism in art, that turns out to be fascinating in its own right, and a real stroke of genius on Hofstadter’s part to serve as one of his analogies for consciousness.

Godel’s theorem did to mathematics and logic what Quantum mechanics did to physics — just as turn-of-the-century physicists cringed when uncertainty and randomness crept into their beloved physics, mathematicians were horrified to learn of Godel’s discovery — basically it did for mathematics and logic what Relativity and Quantum Mechanics did to physics, so it’s really a wonder that it’s not more famous. Godel’s insight, in a nutshell, was that a certain level of uncertainty and randomness exist even in systems of logic. Remember your high school geometry? You start with a handful of given “axioms”, from which you try to prove new theorems (or fail to prove them). Typically you chain together small steps of reasoning (using axioms and other theorems you already proved) and hope you stumble along the right path to emerge, sweating and feverish, at the theorem you were trying to prove.

Around the turn of the century, mathematicians were daring to imagine if this process could somehow be automated, whether you could envision proving all possible theorems derivable from a given set of axioms, by just chaining together every possible series of steps. If you could do this, you could in theory prove *everything* that could be proven — any true statement about any scrap of geometry, for example, could in theory be proven by chaining together the handful of axioms in every possible way. Number theory is another regimented “logical system”, like geometry, where a couple of axioms that are “given” can operate on some simple idealized things — think of the basics of addition, subtraction, etc. operating on whole numbers, which somehow leads to complicated ideas like prime numbers and Fermat’s Theorem. Godel managed to prove, amazingly enough, that there are theorems in number theory that can’t be proven by the axioms of number theory — in other words, he found a mathematical statement that is *true*, and yet completely *unprovable*.

Strange, eh? “How the hell could you ever *prove* something like that???” you yell, sitting in your easy chair in your overalls, waving a spent chicken wing for emphasis. Easy partner, put the gun down, I’ll explain. To horribly oversimplify, Godel found the equivalent of the statement “This sentence is False” within number theory. Of course the trick with “This Sentence is False” is that it’s talking about itself, in fact contradicting itself. Godel was able to construct a theorem made out of the usual math-y bits (you know, greek letters, backwards E, upside down A, the usual) that stated that it could not be proven within number theory. Make sense? No? Well, it took Hofstadter 700 pages, and *he* won a Pulitzer for it, so fat chance I’m going to have.

Hofstadter is fascinated with the idea of a theorem, a bit of logic, talking about *itself*, and furthermore making a bold claim on the abilities of mathematicians to prove it. It’s one of many cases of “strange loops” that he describes so well. And this theorem deserves to be Puliter-ed. It literally crumbled the foundation of mathematics and logic when Godel published it, as it showed that the logical system underpinning number theory (and therefore all of mathematics) contained some truths (i.e. theorems) that were *unprovable*, completely unreachable using the tools built into the system. And furthermore, the same result applies to *any* logical system — doesn’t have to be number theory, it could be *any* collection of axioms and rules that operate on simple elements. That’s what shook the math geeks of the day, that their beloved system had a flaw, an inherent uncertainty to it, and it was built into the system — if they escaped the problem by moving to a completely different system, the “flaw” was still there, since Godel’s theorem still applied. It did to math exactly what the double-punch of quantum mechanics and relativity did to classical physics, introducing inherent messiness into a supposedly pristine crystalline theory.

Here’s where things get weird. As a college physics teacher of mine once said, things that are weird and unexplained often get lumped together, for example the many people who believe quantum mechanics underlies the seeming randomness in human consciousness. You would be forgiven for pondering whether Godel’s theorem is not just an analogy for the human mind, but might actually *apply* to the human mind. After all, the brain is at core a bunch of simple parts behaving under the laws of physics, which you could consider to be a kind of logical system. It’s a pretty attractive idea — could Godel’s Theorem be somehow at play in the brain, perhaps even be responsible for consciousness? Hofstadter’s book spends hundreds of pages on Godel’s Theorem and consciousness — could this be what he’s driving at? You might find yourself, as I did, spending the first 500 pages of the book anticipating that this will be the centerpiece of his Grand Theory, the grand punchline at the end of a grand book.

But no, Hofstadter has no intention of asserting that Godel’s incompleteness theorem actually can be applied to the human mind, instead leaving it solidly in the camp of an analogy to, not a description of, consciousness. It’s a little disappointing to see him arguing against it’s involvement in consciousness, but it’s probably the correct conclusion. Plenty of other people *have* dared to connect the dots, to suggest that Godel’s theorem applies to the human mind, and is responsible for what separates us from stale “soulless” computers. It’s a compelling idea, that since Godel’s theorem suggests there are true statements in logic that cannot be proven, perhaps we (as instantiations of a logical system of sorts) are capable of knowing things that cannot be proven by science or performed by a computer. What’s more, some argue that Godel’s theorem must apply to the laws of physics, since after all, physics at base level is a handful of simple rules acting on elementary objects (and so qualify as a system of axioms). These kinds of grand ideas are more than enough for some pseudo-scientist to run with, perhaps build a whole cult around. But Hofstadter wisely shoots down or ignores these tangents — after all, he says, there is plenty enough power and complexity in a standard-issue brain for consciousness to emerge, without the need to invoke Godel’s theorem or quantum mechanics or any other funny business.

Moreover1, it isn’t clear how to find the true-but-unprovable statements that must exist, other than the simple one that Godel used in his proof, if indeed any others exist at all. It could be (as some logicians believe) that these mysterious statements are all of the same self-referring family as the one Godel used in his proof, and that none of them actually state anything profound or meaningful. I’m not sure if anyone has to date found any other of these true-but-unprovable beasts that the romantically-inclined suggest lurk out in the fringes of mathematics and logic.

Hofstadter is in a category of his own among the scientists I survey, namely the category of “might actually be a correct theory”, by virtue of his *not* espousing that Godel’s theorem actually applies in the brain. It’s a bit of a tease, in my opinion, to go on so heavily about Godel’s theorem if in the end it’s only an analogy for but not applicable to the brain, but I grudgingly accept it because he makes such a compelling analogy. And he’s such a damn good writer.

And that sums it up — a damn good book by a damn good writer. This might in fact be my favorite science book of all, and probably would be one to pack in my desert-island luggage. It’s too bad we can’t all maintain the same enthusiasm for science that Douglas Hofstadter has clearly kept up throughout his career. And, I’m sure you’re thinking, it’s too bad the writers of half-baked book review websites couldn’t be as good a writer as Hofstadter. Point taken. For you, dear reader, all I can do is encourage you to check back at this website now and then in the future. Eventually, TimeBlimp.com will reach sufficient complexity that it will become a self-aware strange loop, at which point it can edit it’s own damn self…


Further Reading:



1.  “Moreover”!  I’ve been dying to use that word.  Another one is “heretofore”.  Has anyone registered heretofore.com?



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