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 never 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. 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.
UPDATE: 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!"
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 never 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. 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.
UPDATE: 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!"
I would prefer that infinity is an strange concept that we use at our peril. One that we can imbue with some properties but also derive much nonsense.
ReplyDeleteSurely the fallacy underlying Zeno's paradox is that an infinite number of identifiable events can only occur in a finite amount of time. I can take the events of a finite time interval (e.g. a person traveling 1 kilometer) and identify an infinite number of sub-events happening in there, but that doesn't mean an infinite amount of time is required. It just means that I have chosen to describe this in a manner that requires an infinite number of lines. ("First he moved from x=0 to x=0.5. "Then he moved from x=0.5 to x=0.75...")
ReplyDeleteBut Alex, Zeno does not discuss the time required at all. He only asks, "How can a task requiring an infinite number of events ever be completed AT ALL, in whatever time we might allow?"
DeleteI've suggested in another post that the real solution is that space is actually chunky (quantized).
And as always, I suggest _Atomism and Its Critics_ for a very thorough investigation of what is being argued in these debates. What is really at the root of this is, is the continuum just a mathematical construct, or does it describe real space-time?
Gene, apart from the time objection, what is the problem with an infinite number of events being completed? Is the problem supposed to be that that contradicts the idea of infinite, which means "never completed"?
DeleteWell, the fact that the series goes on forever?
Delete