I couldn’t come up with 5 varieties like I did with history and theology some weeks ago. In any case, here’s an attempt at delineating modes of interplay available to the fields of philosophy and aesthetics.
Philosophical Aesthetics. Or philosophy of beauty. This is a branch of philosophy. Typical analytical modes of evaluation will be deployed to interrogate the adequacy of the prevailing conceptual apparatus used to frame how we talk and argue about art, beauty, etc. Consider Immanuel Kant’s Critique of Judgment.
Aesthetics of Philosophy. A branch of rhetoric (?) Something like the study of philosophy as literature. Here rhetorical and literary categories of description and evaluation will be deployed to bring to view how the presentation of a philosopher’s authorial voice complements content, projects a model of reasonableness, and cultivates a taste for the charms (and thereby reinforcing the persuasiveness) of its modes of appeal. For instance, consider how the operative aesthetic sensibilities informing Wittgenstein’s Tractatus control his thought and expression—how W.’s prose show us a glimpse of a beauty of which we cannot speak. Related is Clifford Geertz’s Works and Lives, or nearly anything by Wayne Booth.
Aesthetic Contemplation. This is not a branch of philosophy but an approach to the pursuit of truth. Let’s call it imagination seeking understanding. Or perhaps the study of literature/art as a philosophical stimulant. Works more by an appeal to the imagination as the basis of its rhetorical strategy. It isn’t trying to steamroll you by way of its incontrovertible rationality, so it doesn’t wear its logic on its sleeve. Instead it proceeds by exploratory insight and life-like description, so its leverages of persuasion are less coercive and more invitational. Consider Dostoyevsky’s The Brothers Karamazov.