Broadly speaking, "atomism" is the doctrine that the physical world must somehow exist in discrete chunks. If that were not true, we would be faced with paradoxes, for example, as to how motion is ever possible. The most famous of these paradoxes were framed by Zeno of Elea (490-430 BC), who attempted to show the problems of viewing real space as though it were a continuum. This chapter explores those paradoxes, and some misunderstandings of their import.

Why the theory of limits does not solve Zeno's paradoxes

Zeno noted that in moving from point A to point B, first you have to move halfway to B. Then you have to move three quarters of the way to B. Then 7/8 of the way to B. And so on, in an infinite series. And he wondered how anyone can ever complete an infinite series of moves.

What the mathematical theory of limits shows is that, if one "completes" that series, one will be at point B. Well, Zeno already knew that! The theory supplies a formal way of solving what value an infinite series approaches in its limit. That is something completely different from answering Zeno's puzzle over how we can actually complete such an infinite number of moves!

To clarify, when we calculate the limit, what we do is figure out what number the final result would approach, more-and-more closely, if we were to do every addition in an infinite series. We don't actually do the additions, because that would take forever. But the job of a runner trying to cross the continuum between the starting line and the finish line is not to figure out where he would get to if he actually completed the infinite series of moves necessary to reach the finish line. His job is to actually reach the finish line, not to figure out how far away the finish line is!

Zeno was not worrying about specifications or formal systems

Physicist Alex Small once wrote me, contra these thoughts on Zeno, that:

"the fact that we can specify a process via an infinite list of statements does not mean that it is impossible for such a process to happen. There is an implicit assumption that the only physically feasible processes are those with finite specifications in some particular formal system."

But this is mistaking what the Greeks were worried about. Their concern was not with specifications of formal systems. Their concern was with the nature of space. And they were puzzling over whether space, in reality, was infinitely divisible, or was it somehow chunky, or atomic. And some among them noted that, if it is infinitely divisible, that seems to create some problems, such as it seemingly making it impossible for things to get moving.

The difference between worrying over this and worrying over specifications in formal systems might be clarified by my stating that I have no quarrel with the mathematical concept of the continuum at all.

The fact that in a formal system, something might be specified as taking an infinite number of steps, and that we then treat those steps as if they were completed, leaves me as serenely unperturbed as the Buddha under the bodhi tree. We can have a model of space as a continuum, and if it proves useful, well, for a model, that's all that counts.

The issue here is not about our specifications or any formal system: it is about reality.

The Greek Bay Packers

The New York Giants were inside the Green Bay five-yard line, maybe at around the two. The Packers jumped offside. Normally, that would be a five-yard penalty, but inside the defensive team's five, it's half the distance to the goal line. The teams line up again. Green Bay jumps offside again. Half the distance.

"Hey," I said to my friend Sandy, with whom I was watching the game, "the Packers are implementing the Zeno strategy!" I figured they knew the Giants were going to score and take the lead unless they did something desperate, so they intended to continue jumping offside forever, allowing the Giants ever closer to the goal line but never able to cross it.

That whimsy brought up an interesting point: What could be done about a team really trying to implement such a strategy? Say, there's twenty seconds left in a game, and the team trailing by four has a first-and-inches to the goal line. The defensive team figures there's no way to stop each of four successive QB sneaks from getting the ball in the endzone, in which case they lose. And so they jump offsides the first moment the offensive team lines up, intending to do so forever -- meaning, until everyone gives up.

What can the officials do about this? Is there a response available within the current rules?

More realistically, what if a team just has a tired offensive unit it wishes to rest for a while, and a desire to flummox their opponent. On x-and-inches to the goal, why not jump offsides 10 or 20 times in a row? Certainly, the odds of giving up a touchdown are only changed minutely by a series of minuscule advances towards the goal line.

By the way, the Wikipedia entry is excellent in explaining why the notion of a limit from calculus does not solve Zeno's paradox:

A suggested problem with using calculus and mathematical series to try to solve Zeno's paradoxes is that these solutions miss the point. To be precise, while these kinds of solutions specify the limit point of infinite series, they do not explain how such a series can actually ever be completed and the limit point be reached. Thus, calculus and mathematical series can be used to predict where and when Achilles will overtake the tortoise, assuming that the infinite sequence of events as laid out in the argument ever comes to an end. However, the problem lies exactly with that assumption, as Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, an infinite number of physical events need to take place, which seems to be impossible in and of itself, independent of how much time such an act would require if it could actually be done.

Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned. To be precise, Zeno started with the assumption that a finite interval can be split into infinitely many parts, and then argued that it is impossible to move through such a landscape. For calculus and other series-based solutions to make the point that the sum of infinitely many terms can add up to a finite amount therefore merely confirms Zeno's assumption about the landscape (geometry) of space, but does nothing to answer Zeno's question of how we can actually (dynamically) move through such a space." (This text seems to have been removed from Wikipedia, although it can still be found on some other sites.)

Testing the Limits of My Patience

To rehearse Zeno's "runner's paradox" briefly: A runner is faced with the task of covering the distance between the starting and finishing lines. We can simply designate that distance as one. (One what? Well, one "race distance.") To cover that distance of one, the runner must first cover one half the distance from the start to the finish. Having done that, he next must cover one half of the remaining distance, or one quarter of the original distance. Having done that, he next must cover one half the remaining distance again, or 1/8 of the original distance. So the runner must "complete" the infinite series 1/2 + 1/4 + 1/8 + 1/16... before reaching the finish.

In modern mathematical terms, we talk about "limits," and we find the limit of this infinite series, and see that it is equal to one. Does this solve Zeno's paradox? Clearly it does not:

"A word of caution is necessary, however: the expression lim (n --> inf) 1/n = 0 only says that the limit of 1/n as n approaches infinity is zero; it does not say that 1/n itself will ever be equal to 0 -- in fact, it will not. This is the very essence of the limit concept: a sequence of numbers can approach a limit as closely as we please, but it will never actually reach it." -- Eli Maor, e: The Story of a Number, p. 29

That pretty much settles that: the notion of a limit actually expresses Zeno's paradox, rather than solving it. The runner can get as close to the finish line as we please, but can never actually reach it.

So imagine my surprise to find Maor, a few pages later, claiming:

"It is easy to explain the runner's paradox using the limit concept. We take the line segment AB to be of unit length, then the total distance of the runner must cover is given by the infinite geometric series 1/2 + 1/4 + 1/8 + 1/16... this series has the property that no matter how many terms we add, its sum will never reach 1, let alone exceed 1; but we can make the salmon get as close to 1 as we please simply by adding more and more terms. We say that the series converges to 1, or has the limit 1, as the number of terms tends to infinity. Thus the runner will cover a distance of exactly one unit... and the paradox is settled." -- p. 46

I admit I am flabbergasted: Maor is saying "So you are puzzled as to how the runner ever actually reaches the finish line (one)? Well, see this mathematical process that also never actually reaches the finish line? Right? Well, that explains it!"

Zeno knew that runners actually finish races, and that things actually move around (at least in the world of appearances). What he was pointing out is that there is something fishy about the mathematical idea of the continuum if we try to apply it to space in the real world. And I think the clear way to "settle" the paradox is not to fatuously point to a mathematical process that never reaches the finish line, and say it explains how the runner does reach the finish line, but to recognize that real space must not be a continuum. It is chunky, or, if you will, quantized. And real motion, although, like a movie, it may appear to be continuous, actually occurs in quantum leaps. That, my friends, actually gets around the paradox.

Philosopher Francis Moorcroft makes the same point as I did above:

"This reply, however, misunderstands what modern mathematics has shown. Mathematicians do use sequences such as 1/2 + 1/4 + 1/8 + 1/16 + ... but they say that they have a limit of 1, or tend to 1. That is, we can get nearer and nearer towards 1 by adding on more and more members of the sequence, but not actually arrive at 1 -- this would be impossible because we are considering an infinite sequence. So far from providing an argument against Zeno, mathematics is actually agreeing with him!" (https://web.archive.org/web/20100418141459id_/http://www.philosophers.co.uk/cafe/paradox5.htm)

The Brian Wilson Paradox

"Happy times together we've been spending
I wish that every kiss was never-ending
Wouldn't it be nice?"

My daughter Emma, upon hearing those lines, asked, "But Dad, if every kiss was never-ending, wouldn't that mean there could only be one kiss?"

"Hey, that's right," I responded.

To which my son Eamon replied, "No, there'd be zero kisses."

"Huh?"

"Well, he'd never finish a kiss. And what's more, he'd never even really get any fraction of the way into a kiss."

(His point here being, by the way, a bit different than the Zeno paradox -- he wasn't saying you could never cover any finite time of kissing, but that no finite amount is ever more than 0% of an infinite kiss. Similar problem: Throwing a dart at the real number line from 0 to 1, what is the probability you'll hit 2/3? Answer: 0!)

Discussing this with Wabulon, he noted that there are (theoretical) super-Turing machines that can do x amount of processing in time t, then x more in time t/2, then x more in t/4, and so on, thus completing and infinite amount of time.

"Good point," I said, "but given that Brian Wilson couldn't even get out of bed for around 20 years, I don't think he is an instantiation of one of those machines.

But, thinking it over a bit more, we figured out a way that Wilson could make his kiss never-ending for the rest of us, while it would end for him.

What did we come up with?

Motion Debater Light

I've got a "motion detector" floodlight over my porch door in Pennsylvania. What it actually does these days is either refuse to come on at all, or shine for a week straight, day or night.

I've decided it's my friend Wabulon's fault.

What I believed happened is this: Wabulon was sitting out on the porch reading Atomism and Its Critics, somewhere in the section on Zeno, Epicurus, Lucretius and so on debating motion. The light started reading along, and what it's doing lately is mulling over these positions. One week, it decides, "Parmenides is right! Motion is just an illusion. I refuse to turn on at all." The next week, he buys into Epicureanism and decides that everything is always in motion, and stays on all week.