Zeno of Elea (c. 490-430 B.C.) Pre-Socratic philosopher and disciple of Parmenides. Zeno argued that motion, change, and plurality are logical absurdities and that only an unchanging being is real. His four arguments against motion (Zeno's paradoxes) attempted to demonstrate logically that the notions of time and motion are erroneous.

Probably the most famous of Zeno's "paradoxes" involves Achilles and the tortoise, who run a race. Achilles, being confident of victory, gives the tortoise a head start. Zeno “proves” that Achilles can never overtake the tortoise. His argument goes something like this:

Before Achilles can overtake the tortoise, he must reach the point A where the tortoise started. But then the tortoise has crawled to point B. Achilles must now run to point B. But the tortoise has gone to point C, etc. Achilles gets closer and closer to the tortoise, but never catches him.

This is really just a restatement of his Dichotomy paradox, which goes something like this:

Before a moving object can travel a certain distance, it must travel half that distance. Before it can travel half the distance it must travel 1/4 the distance, etc. This sequence goes on forever. Therefore, it seems that the original distance cannot be traveled, and motion is impossible.

These paradoxes are not real paradoxes, as clearly motion does take place and not very much math is required to show that objects can have motion.

However, what is interesting, is how they came to be seen as paradoxes, and what fallacious thought processes are responsible for the seeming paradox. There are two sides to this situation, the physical and the mental. We are familiar with what happens in the physical universe, and when travelling to some destination, we do go through the halfway points to that destination, and despite Zeno’s assertions of impossibility, eventually arrive. So our mental representation does not match up with what is really happening in the physical universe. Mentally, Zeno’s augment seems ok – so where is the error? It is apparently in our definition of position; in thought, for most of us, our concept of the boundary surfaces of objects are absolute – like in geometry, where a position can be a point with no dimension – zero and infinity can exist with no problem. By definition, that which is infinitely small cannot be reached, and is not be there anyway. The physical universe however, apparently does not have these absolutes. According to modern physics, the physical universe has a “graininess” and so can’t be divided into infinitely small parts, the smallest size being the Planck length[1], there is also a similar graininess in time known as the Planck time[2].

The error then, is taking data from one universe, that of ideas and language and trying to apply it to a different universe which has a different set of laws, that of the physical. It is interesting to note that science did not make very much progress until Aristotle’s teleological[3] thinking was taken out of it, which was in effect a similar separation of the semantic and physical that makes sense of Zeno.


Comments & suggestions Welcome.



1] The Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. This is the ‘quantum of length’, the smallest measurement of length with any meaning. And is roughly equal to 1.6 x 10-35 m or about 10-20 times the size of a proton.

2] The Planck time is the time it would take a photon travelling at the speed of light to across a distance equal to the Planck length. This is the ‘quantum of time’, the smallest measurement of time that has any meaning, and is equal to 10-43 seconds. No smaller division of time has any meaning. With in the framework of the laws of physics as we understand them today, we can say only that the universe came into existence when it already had an age of 10-43 seconds.

3] Aristotle's "four causes" stood at the heart of Western rationality and Western science for a very long time., Aristotle says that in order to know a thing one must be able to answer four questions - The four causes: a.) the material cause: the matter out of which a thing is made (clay is the material cause of a bowl); b.) the formal cause: the pattern, model, or structure upon which a thing is made (the formal cause of a bowl is "bowl-shaped"; the formal cause of a human is "human-shaped"); c.) the efficient cause: the means or agency by which a thing comes into existence (a potter is the efficient cause of a bowl); d.) the final cause(in Greek, telos ) : the goal or purpose of a thing, its function or potential (holding cereal and milk is the final cause of a bowl). The final cause is the most unscientific, but is far and away the most important "cause" of a thing as far as Aristotle was concerned. Aristotle's analysis of phenomenon and change, then, is fundamentally teleological.