It says everything you need to know about the importance of Socrates in Greek philosophy that the period preceding him is known as the pre-Socratic. The next sections will cover the eminent philosophers of this period. The first is Pythagoras (yes that Pythagoras). Russell provides a rather double-edged compliant; “[he] was one of the most intellectually important men that ever lived, both when he was wise and when he was unwise.” It seems that both his successes and his mistakes were influential. Pythagoras established himself in the city of Croton in southern Italy in the late 6th century. There have always been two sides to Pythagoras, and his history is full of mythology. He is known to have founded a religion heavily influenced by Orphism and was surrounded by disciples who claimed he could perform miracles (he described himself as semi-divine). On the other hand he is credited with inventing mathematics in the sense of demonstrating deductive argument and founded a school of mathematics in Croton.
Pythagoras held the Orphic view that souls are immortal; when living things die the soul is reborn in new life in an endless cycle. He also founded a society where men and women had equal rights, all things were held in common, and even intellectual discoveries were owned by the collective. For Pythagoras, the endless cycle could be broken by intellectual contemplation, which brought one closest to God. For him this contemplation took the form of mathematics, which appeared certain and exact. It was therefore superior to everyday empirical knowledge, in the same way that a geometrically described circle is perfect and a drawn circle never will be. The idea that thought is superior to observation would influence Plato significantly. According to Russell, it was also a source of much that was wrong in later metaphysics and theory of knowledge.
The greatest discovery of Pythagoras (or one of his disciples) was that the sum of the length of the sides adjacent to the right angle on a triangle squared is equal to the square of the remaining side, or hypotenuse. Key was the discovery of a proof for the theory. The influence of geometry on philosophy and the scientific method was significant. It was from geometry that the Greeks derived the method for starting with a truth which appeared to be self-evident, and then arriving at theorems through deductive reasoning which were relevant to the real world. In other words, it was possible to discover things about the actual world by noticing what is self-evident and using deductive reasoning. However, the inability to use inductive reasoning (i.e. to generate knowledge from observation) has held back society up until the modern era.
The philosopher Heraclitus lived in Ionia and was prominent at around 500 B.C. His most important doctrine was that things are formed by opposites, which are in constant tension and strife, but through a cosmic justice remain in balance (which sounds similar to Anaximander). This constant strife creates perpetual change in the world, where nothing is permanent. Russell describes Heraclitus’ ethics as a proud ascetism. He believes the soul is made of both fire and water, which are of course opposites. A predominance of fire in the soul makes one noble and wise, whereas a man with a ‘moist’ soul is like a drunkard, stumbling around and not knowing where he is going. Heraclitus believes fire to be the primordial element.
Ancient Greek philosophy can be a world of extremes. Where Heraclitus said everything changes, Parmenides said nothing changes. Neither did he think in terms of opposites. Rather than hot and cold, and light and dark, there are different levels of temperature and light. Parmenides lived in southern Italy and was prominent around the middle of the 5th century. He considered the senses deceptive, and that the world of everyday things is an illusion. He is credited with inventing metaphysics based on logic, which influenced subsequent metaphysicians up to and including Hegel in the 19th century. His metaphysics is essential as follows; everything that we can think and speak of must exist, and since we can think of things at any time, everything must always have existed. This means that things cannot start or end, and therefore that there is no change. This according to Russell is the first example in philosophy of an argument originating from thought and words applied to the world at large. Clearly it has significant issues, but there are kernels of truth in it.
One issue is that we can think of things which are imaginary, such as unicorns. Parmenides might have countered that even though unicorns do not physically exist (as far as we know), the idea of them exists, therefore his theory still stands. Another issue is there are things that self-evidently no longer exist, such as people who have passed away. However, it could be said that we are not thinking of the individual as such, but our idea of the individual, either from our own memory of them or of what we know about them from other sources. However, it is hard to argue that as a society our memory of past people never changes. It is also hard to argue that a person has always existed, even before they were born. Subsequent philosophers did not generally accept the idea that nothing changes, but the speculations of Parmenides led to the view that substance itself is indestructible (a view that was only disproven by Einstein, who showed that matter could be converted to energy).