In Greek mythology, Achilles was a demigod who had a sea nymph for a mother (Thetis) and a mortal father (Peleus). He is one of many characters from mythology who may well have some historical basis. According to one legend, Thetis sought to give him immortality (a character trait of deities) by bathing him in the river Styx, whose waters held the property of making invulnerable those washed in it. Unfortunately, in dipping the child in she held him by a foot which,consequently, was not immersed. This made him subsequently vulnerable in his heel.
In the Apology, Socrates refers to Achilles' activity in the Trojan War in regards to avenging the death of his comrade and lover Patroclus. He knew, as a result of his mother's prophesy, that he was fated to die after Hector, who had killed Patroclus. Despite all this he slew Hector anyway, honor being more important than his own life. Socrates' point about Achilles is that he pursued what was right even though he knew it would lead to his own death. Socrates, then, is a kind of philosophical Achilles, asking and probing the ignorant even though it places him in mortal danger as a result.