Zeno of Elea (c. 490-c. 430 BCE)

Zeno of Elea (southern Italy) was a student of Parmenides, who argued that what is real is One, not many. A consequence of this rational position (“being is, non-being is not”) is that motion, or change, is an illusion (the Way of Opinion, not the Way of Truth). The consequences of this view affect the possible views of space and time, since motion is generally considered as a change with regard to space over time. Zeno constructed a number of arguments in defense of his master Parmenides; some are measure paradoxes, directed against plurality. Probably his most famous paradoxes are four paradoxes of motion, which all argue that motion is impossible (at least given any accounts that had been put forward regarding the nature of space, time, and motion). Parmenides had come in for a great deal of criticism of his views on the One; Zeno is at least deflecting some of the attacks by drawing out the apparent consequences of opposing views and using Parmenides' own rational principle—that whatever leads to a contradiction must be false—against the attackers. Zeno constructed four paradoxes of motion. At least two of them clearly examine the consequences of the suppositions that (a) space and time are both continuous, or (b) space and time are both discrete. One might wonder why Zeno put forward exactly four paradoxes. Because time can be either continuous or discrete, and space also can be either continuous or discrete, that can lead to four possible combinations: space and time both continuous, space continuous and time discrete, space discrete and time continuous, or both space and time discrete. The first and the fourth paradoxes clearly fit the first and fourth combination possibilities. This entry examines whether the middle two paradoxes might be construed as representing the other two options.

The Bisection Paradox

The bisection paradox (sometimes referred to as the “dichotomy”) looks at the consequences if both space and time are viewed as continuous (infinitely divisible). Let us imagine that someone wants to travel from A to B. To get to B, first she must travel halfway; then to get from there to B, she must travel half the remaining distance; and so on ad infinitum. Thus she can never complete the journey from A to B because there is always half the remaining distance left to traverse. If one were in a football game, and the defense kept committing fouls that cost them penalties of half the remaining distance to the goal, the offense could never actually score on those penalties, no matter how many of them there were. The path from A to B is divided as follows:

To get from A to B is an infinite (endless task), because it amounts to passing through an infinite sequence of points (1/2, 3/4, 7/8, 15/16, …) for which there is no last point, so the “end” of the sequence can never be reached.