Philosopher Zeno discovered a set of paradoxes as an attack on Pythagorean mathematics. While his work was lost, second-hand sources show us some of the paradoxes he revealed.

The Apple

Cut an apple in half. Cut the halves in half, and the halves of each half in half, etc. According to Pythagorean mathematics, you can do this until you have an infinite number of slices (i. e. forever). If the pieces have mass and you put them back together, you have an apple that is infinitely large because you have an infinite number of pieces. However, if the pieces do not have mass, the apple has no mass.

Line Segments

This paradox is based on the paradox of the apple. A line segment can be divided into an infinate number of points. If points have length, the segment is infinately long. This cannot be true because the definition of a point states that it has no length. However, if a segment is composed of points that have no length, the line itself has no length.

The Dichotomy Paradox

If someone wishes to travel a distance, they must first travel half. Then they must travel half of that. Then they must travel half of that, etc. It is clear that once they travel half of the remaining distance, the other half always remains to be traveled. Thus, the person can never reach their destination.

Achilles and the Tortoise

This paradox is based on the Dichotomy Paradox. Say Achilles and a tortoise enter a race. The tortoise is given a head start of 100 meters, and Achilles can run 10 times as fast as the tortoise. By the time Achilles reaches the place the tortoise started, the tortoise has moves 10 meters from that spot. By the time Achilles reaches that point, the tortoise has moved 1 meter from the spot. When Achilles reaches 111 meters, the tortoise is still 0.1 meters ahead of him. When Achilles gets to 111.1 meters, the tortoise is at 111.11. Going on in this fashion, Achilles can never catch up to the tortoise. While this seems strange at first, it soon becomes clear that Zeno is merely calculating the limit as Achilles approaches the tortoise (which is at 111.111111...).