Zeno of Elea was a pupil of Parmenides, and undertook his teacher’s belief that change is an illusion by proposing arguments known as paradoxes, his most famous being Achilles and the Tortoise. In this paradox, it is said that Achilles and the tortoise are racing, but the tortoise has a head start of say, 50 metres. Even though Achilles is running much faster than the tortoise, once Achilles reaches the 50 metre mark, the tortoise would have covered another say, 10 metres, and so will be ahead. Once Achilles reaches this point, the tortoise would have covered more ground, ad infinitum, and Zeno argued that this proves that Achilles could never actually overtake the tortoise because Achilles must always reach where the tortoise has already been, at which point the tortoise will have moved ahead. Another paradox is the Dichotomy paradox, a paradox which says that if you want to walk to a certain point, you must first get halfway there, and before this a quarter of the way, and before this an eighth of the way, ad infinitum, and so one cannot actually reach any point whatsoever-change is an illusion. Many philosophers have attempted to solve these paradoxes. Hans Reichenbach argued that the paradox arises from believing time and space to be separate entities. Diogenes the Cynic simply got up and walked to a certain point instead of trying to refute the paradoxes through words. Herman Weyll proposed that there are only a finite number of distances between two points, rather than an infinite number, and so the paradox is resolved.
Pythagoras lived from c. 570—c. 495 B.C., and is perhaps most famously known for his theorem (a2+b2=c2). He was originally from Samos but moved to Croton to avoid being ruled by a tyrant. There he set up a school of people dedicated to scholarship. He posited that there are three types of men (he used the example of the Olympics)-those who buy and sell, those who compete, and those who look on, and Pythagoras believed that those who look on are the best type of man. The Pythagoreans spent a lot of time studying numbers, and came to conclude that the world is made from numbers, and that reality was founded and based upon numbers. A famous comparison was made that the body is like a musical instrument. Just as a properly tuned and looked-after instrument produces good music, a properly cared for and healthy body produces the means for a good life. Moreover, our bodies, like instruments, can easily become out of tune, and so we should tune ourselves regularly. Moreover, it was a Pythagorean idea that music is a medicine that can heal the soul. Such was the power of music for Pythagoras. As Pythagoras’ ideas developed, two schools developed, the mathēmatikoi (μαθηματικοί meaning “teachers”) and the akousmatikoi (ἀκουσματικοί, meaning “listeners”). The former emphasised mathematical and scientific development, whereas the latter honed in on the religious aspects. Overall, it seems that Pythagorean thought was dominated by mathematics, and it was maths which the Pythagoreans based their outlook and their ideas about the world. We are indebted to Pythagoras for his work on numbers, as well as his ideas about man-ideas which influenced Socrates and Plato.