David Rodeback's Blog

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Thursday, September 21, 2006
It's That Time of the Century

In today's offering you see parts of my personality you might have hoped never to see together: the math geek and the literary scholar, in a bizarre dance with the more familiar political blogger. I prefer to call this erudition, but I'll understand if you consider it some sort of bothersome multiple personality disorder. In any case, read on, and good luck . . .

Looking forward to the turn of a century, people with apocalyptic world views tend to anticipate cataclysms (for good or ill). Looking back, we've wondered for a long time if there is not something inherently unstable about the early years of a century. I'm thinking those thoughts about the twenty-first century, these days, but my explanation will make more sense if we first look back.

Trouble in the Twenty-Oughts

As Western civilization fought and thought its way through the nineteenth century, we knew that certain things were provably, observably, reliably, absolutely true. Math was math, including as it did the unshakable postulates of Euclidean geometry. Parallel lines never meet. Squares and rectangles have exactly four right angles. And so forth. On this foundation was built the solid, indisputable, eternal tower of Newtonian physics -- forces and acceleration, gravity, equal and opposite reactions; the conservation of energy, matter, and momentum; and all that. These were things on which one could rely, no matter what conflicting factions might say or believe about more troublesome and elusive things such as God, human nature, and the proper way to order society.

Then, in the early years of the twentieth century, that which could never crumble . . . did.

I Blame Karl, Nick, Bernie, and Farkas (Cool Name!)

Strangely enough, geometry helped cause the collapse.

In truth, there were signs of future trouble 23 centuries ago, when Euclid himself could not prove his "Parallel Postulate," which says, essentially, that parallel lines never meet. No one proved it after him, either, but not for want of trying. That is one reason we call it a postulate: It is something we believe to be true but cannot prove, which we accept so we can use it to prove other things and explain and control the world around us.

It seems to be the nature of the universe that the act of creating a thing also creates at least the possibility of that thing's opposite. The existence of good includes the possibility of evil. The presence of light creates shadows. And Euclid's invention of Euclidean geometry also created the possibility of non-Euclidean geometry.

German genius Karl Friedrich Gauss saw the clouds beginning to gather early in the nineteenth century. He invented a non-Euclidean geometry but did not publish it, perhaps fearing the depth and breadth of the upheaval it would cause. In 1829 Russian mathematician Nikolai Lobachevsky, whose mentor Johann Christian Marten Bartel was Gauss's friend and former teacher, published a treatise on non-Euclidean (specifically, "hyperbolic") geometry. Three years later Gauss's Hungarian friend Farkas Bolyai published his own treatise based on years of his own, apparently independent work. Then in an 1854 lecture German mathematician Bernhard Riemann founded what would come to be called Riemannian geometry, encompassing and radically broadening his predecessors' non-Euclidean adventures.

But all this is painfully abstract, you say, and you wonder why you're still reading, not to mention what this has to do with politics, the major theme of my little blog. Let me try to make this a little more concrete.

The Nitty-(Geometric)-Gritty

In Euclidean geometry, squares and rectangles have exactly four sides and four right angles, and opposite sides are parallel. The four angles, 90 degrees each, add to 360 degrees. In fact, the sum of the angles of all four-sided two-dimensional figures, whether squares or rectangles or not, is 360 degrees. A triangle has exactly three sides, and the sum of its angles is always 180 degrees. You may not remember this from school, but it sounds very familiar and comfortable, right? It sounds true. In fact, it feels True.

In Lobachevsky's geometry, the sum of the angles of a triangle is less than 180 degrees. There is also a disquieting four-sided figure with exactly three right angles and one acute (less than 90-degree) angle -- which means, depending on how you look at it, that parallel lines meet, or that they don't exist at all.

We tend to envision space as rectangular or cubic. But in Lobachevsky's geometry space curves in one way, and in Reimann's it curves in multiple ways. You might think of longitude lines on a globe, which at the equator appear parallel -- and for all practical, local purposes are parallel in the Euclidean sense -- but which finally intersect at the poles.

It gets weirder, but you get the idea.

You might think these gentlegeeks had been sitting in their ivory towers far too long, eating unrefrigerated, week-old Continental cuisine and smoking their nineteenth-century gym socks. After all, everyone knows that Euclidean geometry is the truth. Not only do we study it in school; it works! We see its proof and its products all around us every day. A straight line is a straight line, and two parallel straight lines never meet, and that's the way it is! And a flat, two-dimensional plane is a flat two-dimensional plane! And the earth itself -- discounting all the mountains and valleys, of course -- is flat, too!

Uh, oops. On second thought . . . Houston, we have a problem. Cognitive dissonance dead ahead, on a collision course . . .

Trickle-Down Trouble (Its Name Is Al)

Back to our history. It took years for all this to filter through the mathematic, scientific, and philosophic communities. By then, we were on the cusp of the twentieth century, and a tropical storm was forming off our intellectual shores. By landfall, Albert would have become at least a Category Five philosophic hurricane. Reimann's work pushed said Albert firmly toward Western philosophy's New Orleans.

I mean Albert Einstein, of course, the Austrian clerk who demonstrated that time itself is relative, that energy and matter are interchangeable, and that Euclid and Newton are not True, but only mostly true. Specifically, Euclid's geometry and Newton's physics do not fit how the universe works when matter, including those pesky subatomic particles, approaches the speed of light, or when we're dealing with distances on a cosmic scale (as in, orders of magnitude larger than my back yard).

Here's the essential point: Einstein's physics, including his theories of relativity, is built upon Reimann's geometry, not Euclid's. You might think this a minor technical point, but it was enough to breach the philosophical dikes.

Postulate for a moment an intelligent, well-educated, deep-thinking person who doesn't accept what much of the world represents as divine absolutes, because he cannot see the Divine, and because said absolutes really don't submit well to proof by human reason (again, not for want of attempts). Now the firmest absolutes in his mind, Euclidean geometry and Newtonian physics, on which the scientific and technological triumphs of modern society are built, prove to be not quite true. Can you see how this person might draw the conclusion that, if not even those things are absolute, there can be, in fact, no absolutes at all? That modern civilization is built at best upon almost-truths?

If It Surfaces in Russian Literature, It Must Be Real

I don't wish to be tedious, so I'll offer only one example of how this upheaval spilled into Western culture at large. (You can add some of your own.) In case you're interested, I'm relying on some of my own unpublished research from my too-brief academic career in the fascinating but economically impractical field of Russian literature.

Enter Boris Bugaev or, as he is more widely known, Andrei Bely (pronounced "byelly," like a hybrid of "belly" and "yell"). The son of a world-class mathematician and himself a mathematician, Bely wrote (among other things) what is widely regarded as one of the world's best twentieth-century novels, Petersburg (1913). Set in Russia on the verge of the 1905 revolution -- the early years of the twentieth century, mind you -- it is the story of a revolutionary who is ordered by his organization's leaders to assassinate a government official: his own father. Political revolution and patricide already undermine essential traditional pillars of human civilization, but there is more.

On another level, St. Peterburg, Russia's quasi-European capital city, is well-ordered, uniformly numbered, and rectilinear. But in Bely's novel it under attack by a very real but abstract force. This invader is elemental chaos, entropy -- represented as spherical or curved space. It's as if Einstein and Reimann are tearing Euclid and Newton to shreds, and all of modern civilization with them.

Back to the Present

So much for the twentieth century. Now we're in the early years of the twenty-first. The world didn't end with the millennium (of which the first year was 2001, not 2000 -- remember that delightful little debate?). The following items seem to fit the pattern of essential chaos in a century's infant years.

  • We're in a world war, using actual guns and bombs, but we're having trouble naming our enemy.
  • We're no longer able to agree on what marriage is and is not, or what a family is.
  • We're finding it increasingly difficult to have an election and agree upon the winner.
  • It's perfectly fine with much of the educational establishment if Derek and Desiree (Jack and Jill's offspring) grow up thinking that 2+2 equals a unicorn.

Finally, in that same spirit, I present the piece -- or planetoid -- de resistance. I returned home from work the other day and saw a sign of the times on the kitchen counter. It was my middle son's prized plastic model of the solar system, which he purchased last year or the year before that from the Scholastic Book Club or some similar establishment.

When he first assembled it, there were nine planets, revolving independently around the sun, each on its little black plastic arm, with sizes and distances understandably not quite to scale. When I saw the model on the kitchen counter the other day, Pluto was removed and lying, still attached to its arm, apart from the rest of the apparatus. You can tell my son pays some attention to the news.

The mind -- well, my mind, anyway -- could readily embrace the discovery of new planets. That's how we got Pluto several decades ago, after all, and the existence of at least one additional planet has been theorized for a long time. There's something inviting and mysterious about the putative Planet X. I could even face the physical destruction of a planet with equanimity. After all, stuff happens.

But according to the news, it is not a case of Pluto having once been a planet, and having ceased to be. No, this is worse: We are told that what for my entire lifetime we have known to be a planet actually was never a planet in the first place. It was merely a "planetoid" or a "minor planet" or a "dwarf planet" all along -- not like the real planets. Like the shattering of the Santa Claus illusion, one feels this on a visceral level, and it feels . . . wrong. Just wrong.

Very wrong indeed.

Such are the times. What will (retroactively) disappear next? Polaris? Alpha Centauri? Neptune? Mars? California? Howard Dean?

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