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If you're looking for the one Big Idea that has most influenced me, it would have to be empiricism—the idea that all knowledge is ultimately grounded in experience. However, considering I am always classified as an empiricist and that I make reference to the importance of experience in pretty much everything that I write, from theoretical philosophy to morality to literary theory, I'm thinking you might just have figured that one out already. Have you heard the news? Hume the empiricist was influenced by empiricism! I'm like, duh.
You wouldn't think this guy would have had much influence on me, since God plays a central role in his thinking, serving as the purported basis of all human cognition. According to Malebranche's doctrine of occasionalism, God is actually held to be the only true causal agent, since there are no actual cause-and-effect relationships between phenomena, whether mental or physical.
I, by contrast, find the claim of God's existence to be a dubious proposition in itself, and I regard any appeal to the divine in a philosophical explanation to be, for lack of a better word, just silly.
And yet, and yet… something in Malebranche grabbed me, which is why I once recommended in a letter to Michael Ramsey that readers of my Treatise would do well to look at some of his writings (source). What was it that I liked? Malebranche's analysis of causation, for one thing. No, I didn't buy his stuff about God being the ultimate cause, and I explicitly rejected the doctrine of occasionalism—but Malebranche did show that we have no way of making sense of a cause as something having a necessary connection to its effect. And that idea is, as you know, absolutely central to my whole critique of causation.
I don't always admit this, but Malebranche also influenced me in my denial of the existence of the self, as well as in formulating my distinction between matters of fact and relations of ideas. So my advice to Ramsey still stands: if you want to understand Hume (and I can think of few worthier aspirations), read some Malebranche.
I know: I listed Descartes as one of my rivals. And he most certainly is—it's not as if I'm changing my mind on that. But I have to acknowledge that in many ways, Descartes's Meditations established the starting point for my own philosophizing. I follow him in accepting the view that philosophy properly begins with epistemology in general and, in particular, with the relation of the mind to its own ideas. So, like Descartes, I begin with the interior and the subjective, and only then do I try to work my way outward.
In accepting this basic framework, I end up facing many of the same questions as Descartes does in the Meditations—questions about the existence and nature of a world outside of my own mind, as well as questions about the nature of the self. But whereas Descartes tries to answer these questions, typically by bringing in God, I am content to show how these problems are unresolvable. I know that not everyone likes that approach, but, hey, at least I'm honest.
There's something else you need to know about us philosophers: we are sort of obsessed by the people we dislike. No, putting it that way makes it sound a bit too creepy. Let's say instead that we are often most inspired by those whose ideas we most want to refute. That sounds a lot more high-minded. So, yes, I've taken certain ideas from Descartes. But in the end, he's more important to me as someone I truly, deeply, madly want to prove wrong—across the board.
If I remember correctly, it's mainly for that reason that I suggested in my letter to Michael Ramsey that reading Descartes's Meditations would also be useful preparation for reading me.