Zeno's Paradoxes

According to the Philosopher Parmenides a thing could only be what it is and couldn't change into something that wasn't there before it. He concluded that the appearance of change is an illusion and that appearances can only produce opinions of the world, not truth.

Zeno was a pupil of Parmenides and set out to defend his teacher by developing some paradoxes.

He demonstrated how misleading the senses can be because one millet seed falls silently to the ground whilst 1,000 seeds makes a sound. This suggests that 1,000 times nothing makes something. According to Zeno we can only use thinking to determine the truth of the matter, and he therefore concluded that it is better to rely on thinking than on our senses.

Now according to Pythagoras everything was divisible into small units. This meant that a sports track was divisible into an infinite number of points and it should be impossible to cover it in a finite or specified amount of time because each point would first have to be crossed before arriving at the next one. This process should proceed indefinitely and take an infinite amount of time. Thus, if a tortoise got a head start in a race with an athlete, the athlete would never be able to pass because the distance between them would always be divisible into an infinite number of points, each one having to be crossed by the athlete!

The paradoxes of Zeno were intended to illustrate the absurdities that arise when things are divided into small units. However, they should not lead us to mistrust our senses because when we go against our sensations we lose the only standard of truth by which we can objectively measure reality. If fact, Zeno's paradoxes illustrate this clearly because irrespective of how confused we might be our common sense and practical experience indicate clearly that we can cover the distance between two points without any difficulty and beat any tortoise in a race. Similarly, when we place a stick into a glass of water our sense of sight doesn't mislead us by giving the stick the appearance of being bent- what it alerts us to is the refraction of light due to the different speed with which light travels through water and air.

The next paradox of Zeno was based on the division of time. According to Zeno, an arrow at rest occupies a space equal to its length. If it is shot at a target it still continues to occupy the same space as its length, and at any point or moment during its flight it should therefore always be at rest.

Once again, our senses tell us that the arrow does move, and this alerts us to the possibility that the error lies with time that has been divided into moments and, in addition, Zeno uses a static definition of motion. (Einstein later showed that any object in motion changes its dimensions although at the speeds we are normally accustomed to these differences are indiscernible.)

The rational solution to Zeno's paradoxes are lie in the assumptions they make. All of them rely on the abstract concept of division which was created by mathematicians to assist them in describing nature. However, usefulness is no criterion of the truth and in nature, or reality, objects are not broken up into units or points- a track or stick is one continuous whole. Similarly, there is no such thing as units or moments of time. In reality time is continuous and flows in a continuous, never-ending stream.

'As far as the laws of mathematics refer to reality they are not certain, as far as they are certain they don't refer to reality.'
Einstein