Ancient Greece (particularly the polis) are fascinating to study. The short answer would be that in these polis (ie. Athens, Corinth, etc) there was no standing army and, as Socrates states, wasn't any military training programs administered by the 'state' (in this passage that word is likely very loosely translated and certainly shouldn't be understood as we understand the word today). When Athens, for example, went to war with another polis or 'state', the able men of the city would prepare for war as there was no standing army or even established generals. The Athenians (and others) fought as they lived - together, as a unit. The only unique war culture (in relative opposition to much of the polis culture) in ancient Greece existed in the Macedonian kingdom around the Hellenistic era. The Macedonians were ruled by kings and these kings, as well as any others wanting to gain respect, would need to 'proove themselves' in battle, preforming 'heroic acts' in order to become powerful and respected.