There is a burgeoning field within philosophy known as computational philosophy. This term covers a large amount of work being done, including everything from computationally discovering connections between different philosophical topics to artificial intelligence and the philosophy of mind. One fun example is how automated search methods found a connection between a medievals philosophical idea and a modern one. (In this case, the weird but fascinating outdated idea of "divine illumination" is connected to "mental representation" through "the medievals problem of universals.")

One of my favorite examples of computational philosophy is in metaphysics. During the Middle Ages, it became fashionable to attempt proofs for God's existence. And the most famous of these (and most logically strange and not compelling) is Anselm's ontological argument for the existence of God. Essentially, the argument revolves around assuming that God is the most perfect being imaginable and since existence is more perfect than non-existence, God must exist (here's the argument in more detail).

Since then, others have come and refined the argument as well as criticized it. But one of the most intriguing variants of this argument comes from computational theology.

In computer science and mathematics, there is a long tradition of computational proof: using a computer program to derive mathematical proofs from a series of axioms. So two philosophers decided to connect this to Anselm. When they set out to have a computer program prove the ontological argument, not only were they successful, they actually made his argument more elegant. As described in Wikipedia:

Paul Oppenheimer and Edward Zalta used an automated theorem prover—Prover9—to validate Anselm's ontological thesis. Prover9 subsequently discovered a simpler, formally valid (if not necessarily sound) ontological argument from a single non-logical premise.

Separate from whether the proof "works," it's quite impressive that a computer program was able to distill this argument to its simplest form, allowing one to examine it as clearly as possible. In this case, the entire argument rests on the premise "if the conceivable thing than which nothing greater is conceivable fails to exist, then something greater than it is conceivable", which the authors find wanting. But as the philosophers wrote in their paper:

Anselm’s ontological argument has come in for criticism ever since it was first proposed. But we think that the focus on finding flaws in the argument may have hindered progress in logically representing the argument in its most elegant form. We hope to show that computational techniques offer a new insight into Anselm’s ontological argument and demonstrate that there is much beauty inherent in its logic.

While in many cases machines are making discoveries that are increasingly complicated, this is not always the case. As seen here, computers can help us find the beauty in our own, sometimes flawed, arguments.