Any Two Points Define a Line Segment
I know this is late, but let’s talk about the Testamento geométrico. It seems to have captured some interest, and I want to push it a bit. As a preliminary matter—even though I’m disinclined to trust any text that tells me something is obvious—Amalfitano appears to be probably correct when he says the book “obviously” came from Santiago de Compostela rather than Santiago de Chile. The phone numbers for the bookstore are plausible phone numbers for the Spanish province of A Coruña (also La Coruña, which are both the names of the province’s capital city as well), where Santiago de Compostela is located. How the book got from there to Santa Teresa remains a mystery, but it is at least a confirmable known unknown.
The three sections of the book are laid out on p. 185: “Introduction to Euclid, Lobachevsky and Riemann,” “The Geometry of Motion,” and “Three Proofs of the V Postulate.” Euclid is of course the Father of Geometry; his Elements is one of the monuments of mathematics. Lobachevsky formulated the first non-Euclidean geometry, in which lines that are parallel are not equidistant from each other at all points. And Riemann formalized nonhomogeneous non-Euclidean geometry (which to my mind—having no formal math training beyond the first rank of college calculus—sounds like a similar-magnitude advance over Lobachevsky to that which Lobachevsky accomplished over Euclid).
The geometry of motion I have no information on; it seems like it might just mean nontransformative movements, the things you may remember from junior high as translation (sliding, and how’s that for a loaded technical term in our discussions?), reflection (flipping), and rotation (spinning). If that’s the case, though, I have no idea how it could possibly be worth an entire section of a book. I basically don’t know what’s in this part, or how to figure it out. The hazards of trying to expatiate on the contents of a nonexistent book.
“V Postulate” I originally read as the letter V, but it’s actually a Roman numeral, and this section of the Testamento thus purports to offer three proofs of the fifth postulate of the Elements. (“Amalfitano had no idea what the V Postulate was or what it consisted of, nor did he mean to find out.” This is what we call a red flag.) The postulate says that if two lines intersect a third line at angles that sum on one side of the third line to less than 180°, when you extend those two lines in the direction of the side where the angles sum to less than 180, those two lines will eventually intersect each other. Seems intuitively obvious, but there’s more to say about it.
Personally, although I’m viscerally repelled by the abuse of a book, I think the idea of teaching a geometry book a thing or two about the real world by hanging it on a clothesline is very funny. (I actually thought much of the Part About Amalfitano was quite funny, while at the same time dread-full.) And people have already remarked on the book’s symbolism of Amalfitano himself, utterly passive with respect to their environments.
This V Postulate thing, though, bears some scrutiny. Like I said, it sounds plainly manifest to the intuition, but it gave geometers and philosophers about two thousand years’ worth of trouble. (Here’s an easy 90-year-old article on the subject.) Partly that’s because of its complexity; the other four postulates are as simple as “All right angles equal one another” and “There is such a thing as a circle.” Because of this complexity, the fifth postulate seems more like a proposition (the items that Euclid proves by doing geometric constructions based on his definitions and postulates), and thus like it should be provable rather than just assumable. But the proofs have been notoriously slippery, using hidden assumptions that amount to rewordings of the very thing they’re trying to prove.
Schopenhauer (in Die Welt als Wille und Vorstellung) thought it was basically stupid to try to prove the fifth postulate, because the necessity of a proof indicated a prioritization of logic and derivation from first principles over direct, sensory impression. The attempts to prove the postulate, though, created some very interesting and useful results. One of the most productive of these attempts was by Lobachevsky, who began (as many did) by supposing the postulate to be false and looking for a resultant contradiction. What he found instead was a wholly consistent geometry that did not function according to the Euclidean rules that were assumed to order the universe.
And here’s where I’m going with this: The V Postulate, which the Testamento seeks to prove, doesn’t seem to be provable. It is, however, a necessary assumption to one of the foundational systems of human understanding of the world. Loosely put, it is an optional rule that, when adopted, yields a highly useful system of convention; when it is discarded, the result is an equally consistent but very different system. In this way, the postulate is like any number of social rules that are not, sensu stricto, necessary but are essential to the orderly and humane functioning of human interaction; there are modes of human interaction that do not follow those rules, and they can be incomprehensible if seen through the lens of those rules. Some of these rules differ from culture to culture (shades of 2666‘s prodding of national identity), like the cabbie’s view of Espinoza and Pelletier as Norton’s pimp. Others seem like they ought to be reasonably panhuman—no killing young women on a whim. I think we’ve seen lots of examples so far in 2666 of this kind of social Jenga, and the various ways human relationships collapse (and the new and unfamiliar shapes they take) when certain fundamental bases are removed: Espinoza and Pelletier beating the cabbie; Edwin Johns and his hand; lots of what happens with Lola; the general atmosphere of Santa Teresa. I’m sure there’s more. How economical of Bolaño, to figure the whole thing in an object that’s already doing multiple duty as a symbol.