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William Blake, The Ancient of Days, 1794 (CC)

From Scientific American:

Who would have thought about God as an apt topic for an essay about mathematics? Don’t worry, the following discussion is still solidly grounded within an intelligible scientific framework. But the question of whether God can be proved mathematically is intriguing. In fact, over the centuries, several mathematicians have repeatedly tried to prove the existence of a divine being. They range from Blaise Pascal and René Descartes (in the 17th century) to Gottfried Wilhelm Leibniz (in the 18th century) to Kurt Gödel (in the 20th century), whose writings on the subject were published as recently as 1987. And probably the most amazing thing: in a preprint study first posted in 2013 an algorithmic proof wizard checked Gödel’s logical chain of reasoning—and found it to be undoubtedly correct. Has mathematics now finally disproved the claims of all atheists?

As you probably already suspect, it has not. Gödel was indeed able to prove that the existence of something, which he defined as divine, necessarily follows from certain assumptions. But whether these assumptions are justified can be called into doubt. For example, if I assume that all cats are tricolored and know that tricolored cats are almost always female, then I can conclude: almost all cats are female. Even if the logical reasoning is correct, this of course does not hold. For the very assumption that all cats are tricolored is false. If one makes statements about observable things in our environment, such as cats, one can verify them by scientific investigations. But if it is about the proof of a divine existence, the matter becomes a little more complicated.

While Leibniz, Descartes and Gödel relied on an ontological proof of God in which they deduced the existence of a divine being from the mere possibility of it by logical inference, Pascal (1623–1662) chose a slightly different approach: he analyzed the problem from the point of view of what might be considered today as game theory and developed the so-called Pascal’s wager.

To do this, he considered two possibilities. First, God exists. Second, God does not exist. Then he examined the consequences of believing or not believing in God after death. If there is a divine being, and one believes in it, one ends up in paradise; otherwise one goes to hell. If, on the other hand, there is no God, nothing else happens—regardless of whether you are religious or not. The best strategy, Pascal contends, is to believe in God. At best, you end up in paradise; in the worst scenario, nothing happens at all. If, on the other hand, you don’t believe, then in the worst case you could end up in hell.

“Can God Be Proved Mathematically?”, Manon Bischoff, Scientific American/Spektrum der Wissenschaft