The notion of the ‘first principle’ is vague and means different ideas. For instance, when Z claims that X is the first principle, it could mean that X is the source of everything in the cosmos; or that everything is made out of X; or and that everything will return to X. We can say that the first principle is the source, destination and constituent of the cosmos that the pre-Socratic philosophers took it to be the basis of doing ‘philosophy’.
Thales (580 BCE) is often called the ‘first philosopher’ or perhaps more like the ‘first scientist’, because natural science was not separate from philosophy until 500 BCE. Thales was the first in a line of thinkers hailed from Miletus called the Milesians. Though we know very little about Thales or his philosophy, he was believed to be the first person to predict an eclipse. Thales is known for his claim that, “the earth rests on water”; and Aristotle interpreted that Thales held water to be the first principle, or arche, of the cosmos.
Anaximander (560 BCE) was a student of Thales. Anaximander begins his theory of the first principle with a chaotic state in which the chaos starts to spin. Heavy things (earth) go to the center, and lighter things (water, air, fire) go to the periphery (EWAF). Heavier things always travel to center of a vortex. Anaximander is considered to have made the first deductive argument in the history of western philosophy. He contends that any of the four elements: earth, water, air and fire (EWAF) cannot be the first principle. His central supposition is that the EWAF are in mutual opposition, i.e. for instance, water and fire cancel each other (a broad generalization based on very limited empirical evidence). Therefore, he concluded that the first principle must be something other than the four elements and ‘unlimited’ or ‘infinite’. He called it Apeiron.
Anaximander’s student, Anaximenes, maintained that air is what everything is made up from, and everything else just either compression (cold and contracted) or dilation (hot and loosely packed) of air. This belief likely derived from observations of precipitation, evaporation and weather patterns. Anaximander didn’t have the conceptual resource to distinguish between the ‘thing’ and ‘properties’, which Anaximenes advanced when he realized that air become hot when dilation happens, the characteristic property of a ‘thing’ is determined by the measure of compression and dilation.
Pythagoras (570 BCE), the father of theoretical geometry, was an ascetic who created some sort of quasi-religious communities. Pythagoras likely developed or inspired his ideas and mathematical abstractions from the land surveying when he travelled to Egypt and was impressed by its geometric accomplishments. He invented geometry as a theoretical, conceptual discipline. Pythagoras argued that the first principle was ‘numbers’. There are two interpretations of this assertion: 1) that numbers compose physical things; and 2) that the order of the laws of the cosmos may be represented mathematically. The first interpretation seems untenable, but the second interpretation sounds relatively more plausible. Interestingly, the Pythagoreans did not have ‘0’, ‘1’ is the point-unit and ‘2’ was the first number. The point-unit has a magnitude but it is indivisible. This framework does not have room to account for the height of an equilateral triangle of side length 2. The height must be somewhere between 1 and 2, so there must be a fraction of 1. The person who found this out was put to death.
Curd, Patricia, “Presocratic Philosophy”, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2012/entries/presocratics/>
Lectures on Ancient and Medieval European History: http://www.historyguide.org/ancient/pre-socratic.html