28 May Zeno’s paradoxes and the fabric of space and time
Zeno’s four paradoxes listed in Aristotle’s Physics are:
The Dichotomy—That a moving object will never reach any given point, because however near it may be, it must always first accomplish a halfway stage, and then the halfway stage of what is left and so on, and this series has no end. Therefore, the object can never reach the end of any given distance.
The Achilles—That the swiftest racer can never overtake the slowest, if the slowest is given any start at all; because the slowest will have passed beyond his starting-point when the swiftest reaches it, and beyond the point he has then reached when the swiftest reaches it and so on….
The Flying Arrow—That it is impossible for a thing to be moving during a period of time, because it is impossible for it to be moving at an indivisible instant.
The Stadium—That half a given period of time is equal to the whole of it; because equal motions must occupy equal times, and yet the time occupied in passing the same number of equal objects varies according as the objects are moving or stationary. The fallacy lies in the assumption that a moving body passes moving and stationary objects with equal velocity.
The flying-arrow paradox concludes that motion is impossible. Zeno pictures an arrow in flight and considers it frozen at a single point in time. He argues that the arrow must be stationary at that instant, and that if it is stationary at that instant then it is stationary at any—and every—instant. Therefore, it does not move at all. This single paradox may bewilder, but the four together release a commotion of absurdities, profoundly questioning our models of reality. Zeno’s paradoxes raise a fundamental question about the universe: Are time and space continuous like an unbroken line, or do they come in discrete units, like a string of beads? It’s a question that even today’s physicists, who are reputed to be closer than ever to a theory of everything, are struggling with.
Zeno’s arguments seem absurd. We know the arrow flies through the air, yet we may have some difficulty in explaining why or how we know. One may argue that the whole notion of fixing a point in time is absurd and that it makes no sense to say that an arrow appears stationary at any point in time. In mathematics, time is a variable that can be fixed by simply declaring it to be some number. We have formulas that tell us where the arrow is at any time t, so if we let t equal some specific time, then we should know the exact spot where the arrow is at that time. Yet this means that our mathematical models of motion, space, and time are merely intellectual constructions built for the convenience of easy calculations, not for the greater purpose of representing the structure of reality. As we came to understand motion through math with greater sophistication, we shed light on Zeno’s paradoxes. But only by solving the ultimate mysteries of time and space can we definitively solve the puzzles that Zeno put forth at the very dawn of science. He was ahead of his time.
At times during the past 2,000 years, his paradoxes were considered nothing more than picky sophisms of logic with little merit for continued discussion. At other times they were considered embarrassments to mathematicians’ investigations of infinity and the continuum; historians tell us that those paradoxes contributed to the Greek abandonment of such investigations. Almost all of what we know about Zeno’s life is speculation, composed from fragments and historical sources written almost a thousand years after his death. We know that he wrote a magnificent book on philosophy that was used as a textbook at Plato’s Academy, but not even the smallest fragment of it has survived. The fifth-century philosopher and mathematician Proclus, our principal source of information about the early history of Greek geometry, tells us that Zeno wrote a book containing forty paradoxes, but that it was stolen before it could be published. The four known paradoxes come to us by way of Aristotle alone. Dozens of major works written by renowned scholars from Plato to Bertrand Russell have pondered the paradoxes.
Zeno of Elea, a “tall and attractive” intellectual revolutionary, was reading from his famous book on philosophy. He had come to Athens from Crotona in southern Italy with his teacher and lover Parmenides to visit Pythadorus in the Ceramicus just outside the city wall and to attend the great festival. His lines of reasoning were terribly confusing; they seemed to rely on language tricks aimed toward the mystifying suggestion that there is only one single thing in this world—the thing he called Being—and that all else is mere appearance. He argued that if a thing can be divided, its divided parts can also be divided and such divisions can continue indefinitely. From this he concluded that change, and hence motion, is not possible. He finished reading, but his audience was confused. Even Socrates was confused. He called out to Zeno. “Zeno, what do you mean? ‘If things are many,’ you say ‘they must be both like and unlike. But that is impossible; unlike things cannot be like, nor like things unlike.’ That is what you say, isn’t it?” “Yes,” replied Zeno.
What Zeno said makes sense. If two things exist, a third must exist to separate them, otherwise there would not be two things, only one. If three things exist, a fourth and fifth must exist to separate the three. To distinguish between A and B there must be a separator C, and to distinguish between A and C there must be another separator D and so on, thus proving that there must be either only one thing in this world or an infinite collection of things. “So,” Socrates continued, “are you giving just one more proof that two things do not exist? Is that what you mean, or am I understanding you wrongly?” “No,” answered Zeno, “you have quite rightly understood the purpose of the whole treatise.” Zeno went on to argue that nothing changes because change would require a becoming and an end to being. “Therefore,” Parmenides said, “the one which is not, not possessing being in any sense, neither ceases to be nor comes to be.” He and Zeno were thinking that something in an act of change must perform that act in time. So change is equivalent to motion; like the arrow that can never leave the bow, change is impossible.