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 Contents
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.

The Point
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.

10 thoughts on “Any Two Points Define a Line Segment

  1. Maria Bustillos February 21, 2010 / 7:51 pm

    Hi Jeff. Not sure whether you knew that the Testamento Geometrico is a real book–it is, though. Two copies on ABE right now. (“Introducción a Euclides, Lobatchevski y Riemann. Los movimientos en Geometría. Tres demostraciones del V postulado”) Also, Duchamp came up with almost exactly the same scheme for exposing a book to the elements as a wedding gift for his sister (the “Unhappy Readymade”) it was called, and his instructions called specifically for a book of geometry, though not this one.

    “… social Jenga,” ha! Loved that bit.

  2. stevebrassawe February 21, 2010 / 11:51 pm

    Jeff, if you will permit, let me say something back to you to check myself on this. Speaking simplistically, let us say that we have come to view the world assuming that certain V Postulates of morality (for lack of a better term) are proven or can be proven. Are you thinking that Bolaño is slowly revealing the world to us as one where these V Postulates of morality (for lack of a better term) are not proven and not assumed. This world makes no sense to us at the outset, but in fact it does nonetheless have a logic of its own. In other words, Bolaño is playing the role of Lobachevsky here.

    If I am on the right track, it seems to me that the world as Bolaño is revealing it to us is one that Schopenhauer might be quite comfortable with. If I am not on the right track, please stop me before I ride off into the horizon with this.

    • Jeff Anderson February 22, 2010 / 12:51 pm

      That’s pretty much exactly it, Steve, except that I wouldn’t say the V Postulates of morality (fun term) we’re coming in with are proven–they can’t be, but they’re useful if they’re assumed to be true anyway. Otherwise, though, yes. Bolaño is showing us I think multiple worlds intersecting, and each of those worlds has discarded some V Postulate of morality or other.

  3. Todd Murry February 22, 2010 / 4:27 pm

    Another aspect, of course, is the fact that the postulate suggests convergence, which is implied even in the number V itself. I’m ass deep in Lost (the TV show) right now, and I can’t help but put this together with the “Everything that Rises Must Converge”/Omega point optimistic view of the universe, which also works with the idea of the preceding “geometry of motion” section (i.e. we are moving to convergence, i.e. there is a point, or singularity, being moved towards, so there is a point to existence – again, just like the V has a literal point). This might be related to the geography of the novel – everything converges in Santa Theresa, which seems to me to be sort of an event horizon for the singularity I’m talking about here (which, again, works with the idea of disappearance and death).

    But the specifics of his reaction points to an existential dread. Amalfitano doesn’t want to know about that postulate. Are we to take it that Lobachevsky (viewed through this symbolic lens) suggests that we assign ourselves a way of looking at the world so that we can fix it with industry and meaning (a point, a convergence)? Is Amalfitano’s mind flirting with a lack of a point to everything?

    It’s funny that as my mind slides over this stuff, I’m almost trying to make diagrams in my head (Euclid:Lobachevsky:Riemann::Aristotle:Spinoza:Sartre) that are similar to the stuff that Amalfitano is jotting down, except that, with him, I think we’re supposed to be responding to the overall idea that his (or any) attempts to order the universe can’t weather the real world (like the book of math/reason on the clothes line) and thus everything makes progressively less sense as the endpoint of reasoning (as far as this book is concerned) is madness. Or something.

    • Jeff Anderson February 24, 2010 / 8:59 pm

      Todd, I suspect you’re right to think the book might be pointing toward, what, futility? I’m starting to get that feeling, and it’s not a nice one.

      I haven’t seen Lost (I know), but I’m glad to see you bring up O’Connor and Teilhard. Prompted by your comment, I spent a very pleasurable couple hours refamiliarizing myself with Dan Simmons’s Hyperion Cantos, and the quick-and-dirty version of Teilhard’s thought that they implicate, and my copy of O’Connor’s collected stories. Thanks!

      Still, although 2666 appears to aim toward a kind of singularity just like The Phenomenon of Man does, the one is the exact (pessimistic) opposite of the other.

  4. Daryl L. L. Houston February 22, 2010 / 9:00 pm

    There’s something I can’t quite assemble into a coherent thought that’s making me want to tie your discovery here to a reference back during the Edwin Johns visit in which he talks about railroad tracks (parallel lines that seem to converge in the distance) and coincidence and then to tie that to the next section, which feints in the direction of being about fate, which in the Johns section is the inverse of coincidence.

    • Jeff Anderson February 25, 2010 / 3:30 pm

      Well now that you’ve pointed me back to Johns’s speech on coincidence, Daryl (nice catch, by the way!), I’m struck by the mounting suggestion of 2666 that he never belonged in the asylum to begin with, that nihilism and “insanity” are the only sane responses to the actual, actually insane world. I’m surprised we haven’t seen any explicit suicides yet in the book.

  5. Daniel February 24, 2010 / 6:13 pm

    Is the V Postulate maybe an allegory for 2666 itself? Imagine:

    2666 is a line.

    Each part of 2666 is part of a longer story, which correspond to the lines that intersect with our initial line – think of the actual five sections in the book as intersection points, geometrically, that lie along lines that go to infinity in both directions.

    We can imagine that these lines are not parallel, and assume, as Euclid did, that they will intersect at some other point, outside of the scope of 2666 itself. (I’m thinking of something like the recovery of the original master in Infinite Jest.)

    Or, we can follow Lobachevsky and Reimann, and say that this might not be the case. They could be parallel and still intersect at a “point at infinity;” they could be non-parallel and still not intersect. Either of these assumptions corresponds to a different, but internally consistent, reading of the facts, analogous to non-Euclidean geometry.

    Evidence that this occurred to Bolaño as well: in the Part about Amalfitano, he writes that the Testamento geométrico:

    eran en realidad tres libros, <>

    was really three books “with their own identity, but functionally related by their common destiny”

    which seems to hint at the books-within-books structure of 2666 as well.

    • Daniel February 24, 2010 / 6:14 pm

      that full quote should be:

      eran en realidad tres libros, “con su propia unidad, pero funcionalmente correlacionados por el destino del conjunto”

      Sorry, I don’t know how to do Euro-style quotes.

      • Jeff Anderson February 25, 2010 / 3:34 pm

        I like this line of thought, and the evidence so far suggests to me that we’re in post-Euclidean territory. More like JOI’s anti-confluential cinema than the reclamation of the master cartridge.

        And wow, that’s actually a really funny joke the narrator makes about the three books’ unity through their common destiny—which was of course to be hung out on a clothesline and exposed to the elements.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s