SATURDAY: We humans do have our higher pursuits!

SATURDAY, JANUARY 4, 2025

Gödel, Escher and Bosh: "What Can I Do For You?"

Once again, it was one of Dylan's questions, during his Christian period. Three albums emerged from that era, including Saved, the 1980 album which included this heartfelt cut.

We're not religious ourselves. Beyond that, we have no cosmological beliefs, beyond the assumption that we humans—at least those of us in the West—don't have the slightest idea who or what or where we are, or how we got wherever we are, or how the realm in which we're found can best be characterized.

Still, "What Can I Do For You?" strikes us as a superb performance of a song which can have strong secular application. A bit of tape we watched yesterday reminded us of the Christian albums, each of which had at least one song we very much liked and admired.

"What Can I Do For You?" was one such song. As our national culture—such as it was—continues to crash and burn. the song is asking a question we ourselves continue to chase.

Then too, there's what Kevin has said.

As we've long noted, we think Kevin Drum's work on lead exposure is the best work we know about in the whole quarter century of online exposition. Plainly, though, he was trying to trigger us when he listed the Hofstadter book among his twenty favorites in this recent street-fighting post.

Why would he want to lash out like that? We have no idea.

He stuck it in at #11, pretending he meant for its inclusion to go unnoticed. To what book do we refer? We refer to this (award-winning) book, a book of close to 800 pages:

Gödel, Escher, Bach: an Eternal Golden Braid. Douglas Hofstadter, 1979.

As if that isn't bad enough, the publisher includes this grabber as a capsule description:

A metaphorical fugue on minds and machines in the spirit of Lewis Carroll.

Ow ow ow ow ow ow ow! You may be getting our point.

At one point, maybe twelve years ago, we actually tried to peruse this famous book. To be fair, the book did win several major awards. The leading authority offers this short account of the world's longest possible book:

Gödel, Escher, Bach

Gödel, Escher, Bach: an Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter.

By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements. It also discusses self-reference and formal rules, isomorphism, what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.

In response to confusion over the book's theme, Hofstadter emphasized that Gödel, Escher, Bach is not about the relationships of mathematics, art, and music, but rather about how cognition emerges from hidden neurological mechanisms. One point in the book presents an analogy about how individual neurons in the brain coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony of ants.

Gödel, Escher, Bach won the Pulitzer Prize for General Nonfiction and the National Book Award for Science Hardcover.

"Confusion over the book's theme?" Where could that have come from?

In fairness, we included the passage about the awards.  That said, here's what the leading authority tells us next about this award-winning book:

Structure

Gödel, Escher, Bach takes the form of interweaving narratives. The main chapters alternate with dialogues between imaginary characters, usually Achilles and the tortoise, first used by Zeno of Elea and later by Lewis Carroll in "What the Tortoise Said to Achilles." These origins are related in the first two dialogues, and later ones introduce new characters such as the Crab. These narratives frequently dip into self-reference and metafiction.

Word play also features prominently in the work. Puns are occasionally used to connect ideas, such as the "Magnificrab, Indeed" with Bach's Magnificat in D; "SHRDLU, Toy of Man's Designing" with Bach's "Jesu, Joy of Man's Desiring"; and "Typographical Number Theory", or "TNT", which inevitably reacts explosively when it attempts to make statements about itself. One dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid and musical varieties), which is titled "Djinn and Tonic". Sometimes word play has no significant connection, such as the dialogue "A Mu Offering", which has no close affinity to Bach's The Musical Offering.

One dialogue in the book is written in the form of a crab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines that double as an answer to a question in the next line. Another is a sloth canon, where one character repeats the lines of another, but slower and negated.

Ow ow ow ow ow ow ow! By now, you surely must know what we mean. 

For ourselves, we'd be slow to assume that the judges who awarded those prizes actually read all the way through this book. Beyond that, its reasoning largely turns on "Russell's Paradox," a formulation which we regard as the upper-end academic clown show of the last century.

With apologies, this arrives early in the 787-page book, mostly on page 17:

Gödel, Escher, Bach: an Eternal Golden Braid

[...]

In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict between the finite and the infinite, and hence a strong sense of paradox. Intuition senses that there is something mathematical involved here. And indeed in our own century a mathematical counterpart was discovered, with the most enormous repercussions. And, just as the Bach and Escher loops appeal to very simple and ancient intuitions—a musical scale, a staircase—so this discovery, by K. Gôdel, of a Strange Loop in mathematical systems has its origins in simple and ancient intuitions. In its absolutely barest form, Godel's discovery involves the translation of an ancient paradox in philosophy into mathematical terms. That paradox is the so-called Epimenides paradox, or liar paradox. Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." A sharper version of the statement is simply "I am lying;" or, "This statement is false." It is that last version which I will usually mean when I speak of the Epimenides paradox. It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think it is true, then it immediately backfires on you and makes you think it is false. But once you've decided it is false, a similar backfiring returns you to the idea that it must be true. Try it!

The Epimenides paradox is a one-step Strange Loop, like Escher's Print Gallery. But how does it have to do with mathematics? That is what Gödel discovered. His idea was to use mathematical reasoning in exploring mathematical reasoning itself. This notion of making mathematics "introspective" proved to be enormously powerful, and perhaps its richest implication was the one Gödel found: Gödel's Incompleteness Theorem. What the Theorem States and how it is proved are two different things. We shall discuss both in quite some detail in this book. 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.

Gödel's Theorem appears as Proposition VI in his 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I." It states:

To every w-consistent recursive class K of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Fig (K) (where v is the free variable of r).

Actually, it was in German, and perhaps you feel that it might as well be in German anyway. So here is a paraphrase in more normal English:

All consistent axiomatic formulations of number theory include undecidable propositions.

This is the pearl.

So much to ridicule, so little time—and we do regard that as a clown car. But in this, the world within which Dylan longed to serve, that's a prime example of the (high academic) business we've chosen.

The sheer folly of "This statement is false" would almost seem to come straight out of the work of the later Wittgenstein. We've run through this folly several times in the past. Today, we won't go there again.

In fairness, this endless book won major awards! On the other hand, we regard it, on its face, as a work of manifest nonsense on an ascending scale.

(This statement is false, the logician once said. But to what statement was he referring? No such statement existed!)

"What can I do for you?" Dylan once asked. We've been chasing the same puzzle too, with no good outcome in sight.

Our national discourse, such as it ever has been, has descended all the way onto the garbage pile. Our journalists continue to refuse to discuss the actual state of play as an apparent madman ascends our ultimate crystal stair.

Our discourse sits on the garbage pile. On the highest end of our academic discourse, big piles of bosh fail to help.

We regard GEB as the world's least coherent, most self-impressed and self-referential book. Admittedly, it won several major awards, and our view could always be wrong, if only in some minor way which has long escaped detection.

For extra credit only: "All consistent axiomatic formulations of number theory include undecidable propositions?"

Is there anything "normal" about that "English?" On what planet? Discuss!