Like Socrates, I’m absolutely passionate about pursuing friendship. Friendship (philia) is difficult to define in Plato’s writings, in part because the Lysis, the only dialogue that attempts a comprehensive understanding of it, is really not about friendship per se; rather, as has been noted by critics such as David Sedley, the Lysis concerns itself with the object of friendship, that is, the thing to or of which someone is a friend. In a relationship between friends, the emphasis would be placed not on the lover (ho philôn), but rather on the beloved (ho philoumenos).
Why is friendship important for Plato? Well, it forms part of the core meaning of the word philosophy (philo-sophia), which is fortuitous for our purposes, since the investigation into philia is really for the sake of defining its object, in this case, sophia (i.e. wisdom). This might be a profitable way to approach the problem of defining Platonic philosophia, since Plato is rather reticent in his writings about what precisely that means. One approach to this problem in the Lysis (212b1-e5) – inevitably abandoned by Socrates for unclear reasons – involves investigation into the possibility that both the subject and object of loving must “become friends” (philoi gignontai) in all cases. We speak this way, of course: when I call Greg Nagy my friend, I do so because I understand – and I assume that he understands as well – that we are friends reciprocally, and it is assumed that I am his friend because I am the object of his philia and because he is the object of my philia. On this “reciprocal” definition of philia, no beloved is a friend to the lover unless the beloved is a lover of him, that is, unless both “become friends”.
But this proposition runs into problems when we imagine that the beloved, the object to which I direct my love, might lack the natural capacity to love in return. For example, I might claim that I am a lover of peanut butter, and this would not be a controversial claim. But, if we grant that the “lover of” an object is a “lover” if and only if his beloved loves him in return, we have a problem, since peanut butter does not have the capacity to love me in return. Indeed, Socrates presents several examples that demonstrate the impossibility of such a definition. Most notable, for our interests, is the example of “wisdom” (sophia): according to this definition, there “are no philosophers unless wisdom loves them in return.” Explicitly, this passage functions within the larger argument of the Lysis to demonstrate the difficulty of arriving at a sufficient categorical understanding of the beloved.