Tuesday, July 6, 2010

Achilles and the Tortoise

Santa Maria del Mar
Santa Maria del Mar, Barcelona, 2004

I'm still grappling with Vasubandhu's Discussion for the Demonstration of Action. I'm starting to suspect that I'm missing too much context to be able to really figure out what he's saying there. It's clearly a part of a grand philosophical discussion between several schools of thought, none of which I understand. The upshot is that I'm picking up occasional flashes of meaning buried in noise, sort of like trying to listen to a radio channel that's fading in and out due to static.

But I'll give it another couple of shots anyway, and then write down whatever little I have managed to glean from it. In the meantime, as I've been searching for something to relate it to, I've returned to one of the first philosophical exercises I remember grappling with: the paradoxes of Zeno of Elea.

Why bring up Zeno when wrestling with Vasubandhu? Because I have a growing feeling that Vasubandhu is somewhat stuck in the very trap Zeno described, although an elegant solution to the paradox is staring him right in the face.

During Zeno's time, there was a major philosophical debate ongoing in Greece regarding the nature of reality. On one side was a group that we could call the Idealists. On the other, a group we'll call the Atomists.

The Idealists traced the lineage of their school of thought to one Parmenides of Elea. Parmenides taught that the universe is, in reality, eternal, unchanging, and indivisible. Nothing can come of nothing, and nothing that exists can stop existing. Our perceptions of it, which include things like motion and change, are illusory. The only way to apprehend this true reality is through philosophical contemplation. Parmenides's best-known follower, Plato, took this several steps further, most famously in his Parable of the Cave: we humans are like prisoners shackled to a wall in a cave, watching the play of shadows on another wall and mistaking that for reality, whereas the true reality lies in the objects casting those shadows—the ideas, or essences, imperfectly embodied in objects in our everyday phenomenal world.

The Atomists, on the other hand, held that "nothing exists apart from atoms, and the void." The two best-known Atomists were one Democritus of Abdera, and his teacher Leucippus. For the Atomists, everything in the Universe consists of aggregations of indivisible, eternal, and unchanging fundamental units of matter, atoms (from a-tomos, non-divisible). Atoms exist in empty space, through which they move, forming and un-forming ever new aggregations. The Atomists held that the best way to understand reality is by observation and experience.

Zeno of Elea fell into the Idealist camp. His paradoxes were intended to show that motion is impossible, and therefore the Atomists must be wrong. The Atomists simply ignored him, on the grounds that motion is plainly observable, hereby demonstrated by this chunk of marble in motion towards your swollen heads, you idiots.

That was probably a mistake, as the Idealists ended up winning the debate and plunging Western philosophy into two millennia of building ever more fantastic cloud cathedrals in a vain quest to apprehend the Really Real world of ideas. Luckily there were always some people capable of pointing out that a cucumber is a cucumber and the whole universe is the whole universe, and proceeding to make some pickles—or build actual cathedrals, some which are rather impressive, and could never have happened if motion was impossible and everything that existed must always have existed.

Nevertheless, Zeno's paradoxes are quite nice brain-teasers, and a few people with too much time on their hands still consider them worth considering. Personally, I find them most interesting as illustrations of one particular cognitive difficulty with which we humans have a tremendously hard time coping.

Zeno wrote up several of them, but one will do for now: Achilles and the tortoise.

Imagine a hundred-yard foot-race between Achilles, the fleetest of foot of all mortals, and an ordinary tortoise.

Since this is obviously something of an uneven match, we'll give the tortoise a head start of, say, ten paces. The athletes step (or crawl) to their starting positions, and someone fires the starting gun. (Can you picture someone in a khiton and sandals with a starting pistol? I can!)

They're off!

Now, the ordinary blockhead would expect that Achilles would catch up to the tortoise in, oh, about eleven paces, tops, and proceed to the finish line before the tortoise has managed to take more than a very slow step or two. Not so, says Zeno, and here's why.

In order to pass the tortoise, Achilles first has to reach the tortoise's starting line, ten paces ahead, right? OK, now, when Achilles has reached that point, the tortoise will have moved forward a bit, right? Right. So, before overtaking the tortoise, Achilles has to reach that point, right? Yup. So, by the time Achilles gets  to that point, the tortoise will have moved forward another bit, right? That means Achilles will have to reach that point before overtaking the tortoise. By the time he gets there, the dastardly tortoise will have moved forward another little bit, so Achilles will have to cover that distance too, etc. ad infinitum.

Infinitum. Infinity. Zeno's triumphant conclusion is that in order to overtake the tortoise, Achilles will have to complete an infinite number of tasks. Since each task takes some non-zero amount of time, however short that time is, completing an infinite number of such tasks will necessarily take an infinite amount of time. Ergo, Achilles will never overtake the tortoise. Democritus and the Atomists are wrong, Parmenides is right, motion is impossible, all is one, everything that exists always existed, and always will. QED, who you gonna believe, me or your lyin' eyes, etc.

If you haven't read anything about the history of Western philosophy, you have no idea how big a headache this has been. Incredibly intelligent people have wasted an inordinate amount of time trying to figure out exactly where Zeno goes wrong, since in every observed foot-race the faster runners do overtake the slower ones. I'm pretty sure someone, somewhere, has actually staged a foot-race between some Greek named Achilles and a poor tortoise spurred forwards with the promise of a nice bit of lettuce. In fact, our team of philosophers only started to get a grip on this kind of thing with Newton and Leibniz, who came up with some really impressive mathematical tools to deal with continuities, and the paradoxes were only finally put to rest—well, more or less—a bit over a hundred years ago, when mathematicians finally came up with methods that made infinite series tractable.

The interesting thing is that there is no satisfactory solution to Zeno's paradoxes in the frame of reference used by the Atomists either. They're dividing the universe into discrete, indivisible, unchanging, and eternal pieces too, which leads to very immediate problems regarding how such inert pieces can interact with each other in any way, let alone do stuff like "move."

I believe this kind of thinking is very deeply built into our cognitional structures. We can only think of things by naming them. When naming them, we designate a part of the universe as a thing, and other parts of the universe as not-a-thing. The thing only exists inasmuch as it relates to the universe which is not-a-thing. In a very immediate and real sense, our universe consists of these divisions we make up.

The Atomists did exactly what Zeno did in his demonstration, only they applied it a bit differently.

The most satisfactory way out of Zeno's paradox—or Thomson's lamp, for that matter, which is a slightly cleverer variant that still puzzles people today—is simply to recognize that they're all mind games. 1

In our situation where Achilles and a tortoise are having a foot-race, Achilles will overtake the tortoise and reach the finish line at a time when the tortoise has barely moved from its starting point. Zeno's division of Achilles' first ten paces into an infinity of tasks is only a conceptual exercise; it's no more, or less, valid than dividing those ten paces into ten tasks, or one, or a million, of identical or different lengths. No matter how we describe the race, we still have Achilles sprinting joyously forward and past the tortoise, as the tortoise starts to take one ponderous step. Sometimes it's useful to consider Achilles, the tortoise, and the race track as separate entities; sometimes to treat the entire race as one event; we can look at each of Achilles' steps separately, or plot his progress on a continuous graph, or just go and have a beer instead. None of it will make any difference to Achilles or the tortoise, or the referee who's probably wondering what the hell he's doing observing such a stupid race as this one.

It's just that if we want to talk about the race, we do need to conceptualize it somehow—without concepts like 'Achilles,' 'tortoise,' or 'race,' we couldn't really say anything about it.

However, Achilles is not 'Achilles,' the tortoise is not 'the tortoise,' and the race is not 'the race.' As obvious as this may seem (or not?), it took Western philosophy over 2000 years to figure this out, and there are still plenty of people who conflate them. No big surprise, since most of the time 'Achilles' really is the functional equivalent of Achilles; in fact, Achilles would probably be very confused if you didn't tell him 'Yassou, Achilles' when bumping into him at the coffee machine in the morning.

So, what got me from Vasubandhu to Zeno? Just this: as far as I can parse him, Vasubandhu seems to be constantly dividing things into indivisible, fundamental units, like the Atomists. He has cittas of cognition, atoms of matter, and loci—kind of like voxels, pixels in three dimensions—of space. I'm puzzled by this, since one of the fundamental ideas in his philosophical system is 'mind-only'—that the phenomenal world of 'Achilles' and 'the tortoise' only exist in the mind, even as the piece of the universe we call Achilles runs merrily along past the piece of the universe we call the tortoise. Why does he treat 'loci,' 'cittas,' and 'atoms' as if they had an ontological existence of their own? On the face of it, this doesn't make much sense, which probably means I'm misunderstanding him. Or perhaps the grip of discriminating thinking really is so strong that even an intellectual giant like him couldn't entirely shake it in his philosophical system, even if he very probably did do so in his meditative practice.

So back I go, to spend more time with the Discussion for the Demonstration of Action.

1In case you looked it up, the solution to the Thomson's lamp paradox is, "nobody can flip a switch that fast, silly." Think about it.


  1. When I think of Zeno, and I do occasionally, what strikes me as his error is a misapplication of measurement. A misuse of conceptual tools. He attempts to measure a finite thing-a certain distance with a potentially infinite idea or construct-time. Like trying to screw a lightbulb in with a hammer. Like trying to measure the time it takes to write this comment with a teaspoon. Although TS Eliot wrote "I have measured out my life with coffee spoons"

  2. That, too, but I think the fundamental fallacy is mistaking the representation for the reality. You can represent the footrace any way you like, with or without any number of ways of measuring it. However, these representations only affect our cognition of the bit of the space-time continuum that constitutes "the race."

  3. I'm not sure about this solution to Achilles and the Tortoise. You're pretty much just ignoring it. It's like solving a jigsaw puzzle by saying, "It's just a bunch of pieces of cut-up paper!"

  4. What if it turns out the pieces in your puzzle are a random mix from a dozen different ones?