Zeno's paradoxes are a famous set of philosophical problems generally understood to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC). The argument revolves around notions of motion, infinity, and the division of time and space.
Achilles and the Tortoise:
In this paradox, the swift Achilles races a tortoise, who is given a head start. Zeno argues that Achilles will never overtake the tortoise, since whenever Achilles reaches where the tortoise was, the tortoise has already moved ahead, perpetually maintaining a lead.
This paradox argues that to reach a destination, one must first reach the halfway point, then the halfway point of the remaining distance, and so on, ad infinitum. Zeno asserts this requires completing an infinite number of tasks, which is impossible, thereby implying that any motion is an illusion.
Zeno declares that an arrow in flight is at each instant motionless, since it's either at the position where it is or it's not. Thus, if every instant involves no motion and time comprises these instants, motion doesn't exist.
The Stadium Paradox
Also referred to as the "Racetrack" paradox, this puzzle is a detailed consideration of motion in relation to time and distance. In this paradox, three rows of runners are involved, with the middle row moving in one direction and the other two moving in the opposite direction. The paradox proposes that a runner from the middle row, moving to the right, will pass two runners from the other rows in the same amount of time it takes him to pass one.
In more detail, if we take the runner from the middle row (Row B) and a runner from one of the outer rows (Row A), when they start, they are side by side. At the same time, the runner from the other outer row (Row C) is already side by side with another runner from the middle row. Now, if Row B is moving to the right and Rows A and C are moving to the left, by the time the runner from the middle row has reached the next runner from Row A, he has passed two runners from Row C.
This leads to an apparent contradiction: in the same time, the runner from the middle row passes one runner from Row A and two runners from Row C, which suggests two distinct velocities for the same runner in the same time period, contradicting the common-sense view of velocity.
The Moving Rows Paradox
Also known as the "Paradox of Plurality", this paradox is not so much about motion as it is about the nature of plurality and infinity. It proposes that a single unit cannot be made up of an infinite number of parts, as it would then be infinite itself. This paradox challenges the concept of an object being composed of an infinite number of "points" or "moments".
The paradox arises when we think about splitting an object or a unit of time into an infinite number of parts. If a unit is made up of an infinite number of parts, each of those parts would have to be infinitesimally small, or even non-existent, for the unit as a whole to maintain its finite size or duration. However, if each part is non-existent, then how could an accumulation of these parts form a unit? Conversely, if each part does have some size or duration, then an infinite number of them should form an infinite unit, which contradicts our experience.
Several solutions to these paradoxes have been proposed throughout history, such as the practical demonstration by Diogenes the Cynic who simply got up and walked to disprove Zeno's conclusion. Notably, Aristotle proposed that as distance decreases, the time needed to cover those distances also decreases. Additionally, he differentiated between "things infinite in respect of divisibility" and things that are infinite in extension. Thomas Aquinas furthered this point by explaining that instants are not parts of time.
Archimedes developed a method for deriving a finite sum from infinitely many terms that get progressively smaller, and Bertrand Russell introduced the "at-at theory of motion," suggesting motion is just change in position over time. Several modern thinkers such as Nick Huggett, Peter Lynds, and Hermann Weyl have proposed their theories to resolve Zeno's paradoxes.
In modern times, while the mathematical implications of Zeno's paradoxes have been largely resolved with the development of the epsilon-delta formulation of calculus, philosophers argue that not all issues raised by the paradoxes are resolved. Zeno's paradoxes continue to spur debate about the nature of infinity, time, space, and motion.
The paradoxes also have counterparts in other traditions, like ancient Chinese philosophers from the Mohist School of Names who developed paradoxes similar to Zeno’s. The Quantum Zeno effect in modern physics, first theorized in 1958, also parallels Zeno's arrow paradox, illustrating how observation can hinder or inhibit the motion of a quantum system.
In verification and design of timed and hybrid systems, Zeno behaviour refers to a system that includes an infinite number of discrete steps in a finite amount of time. In system models, these behaviours are often excluded since they cannot be implemented with a digital controller.