The ontological argument for the existence of God is one of the most fascinating and disputed arguments in the history of philosophy. Anselm of Canterbury formulated the first clear version in the Proslogion in the eleventh century: God is defined as “that than which nothing greater can be thought.” The core of the argument is that if God existed only in thought and not in reality, then one could think of something greater, namely a being that also exists in reality. But that is contradictory, since God is, by definition, the greatest conceivable being. Therefore God must exist in reality.

God as a priori: the parallel to mathematics

The ontological argument is unique among arguments for God’s existence because it is entirely a priori: the conclusion is supposed to be reached through conceptual analysis alone, without appeal to empirical evidence. In this respect it resembles mathematics. The claim that “all triangles have three sides” is true by definition; we do not need to measure triangles in the world to know it. Godel, who was deeply interested in the argument, drew precisely this parallel: just as mathematical theorems are proven a priori from axioms, he thought God’s existence could be proven from conceptual premises alone. Mathematical truths do not depend on experience; they are discovered rather than constructed. The question, then, is whether God, rightly understood, has a similar status.

Infinity as a logically coherent magnitude

A decisive question for the argument is whether the concept of God, understood as an infinitely perfect being, is logically coherent. Leibniz pointed out that the argument strictly proves only this: if God is possible, then God exists. To close that gap, he argued that the concept of God contains no contradiction, because all divine attributes are purely positive. Infinity is central here. In mathematics, infinity is not a contradiction but a well-grounded logical magnitude. Georg Cantor even showed that there are different orders of infinity. If reason can accept actual infinities as coherent mathematical objects, then there is no obvious logical reason to reject the concept of an infinite God as incoherent.

The infinite as a bridge to reality

Here lies the argument’s most interesting core. If God is infinite, and if infinity is a logically coherent magnitude, then one may argue that God’s nature requires existence in a way ordinary finite things do not. A finite being such as a unicorn can be conceived as non-existent without contradiction. But an infinite being that, by definition, exists necessarily in all possible worlds, as the modal ontological argument claims, cannot be absent without contradiction. This is analogous to the way mathematical truths “exist” more necessarily than contingent empirical facts: they hold in all possible situations. God’s infinity therefore introduces a kind of logical necessity that moves the concept from what is merely mental toward what is real, just as mathematical axioms are not arbitrary inventions but necessary structures of thought itself.

Objections and replies

Kant objects that existence is not a predicate. One does not enrich a concept by adding the claim that it exists. Russell sharpened the point further: existence says something about the world, not about concepts. This objection has force against Descartes’s version of the argument, but it does not fully strike Anselm’s or the modal version. Even without treating existence as an ordinary predicate, one may still claim that necessary existence distinguishes kinds of beings.

Another objection is Gaunilo’s island parody: apparently one could prove the existence of a perfect island in the same way. The answer is that islands, unlike God, do not possess an intrinsic maximum. For any island, one can always imagine one more palm tree, one more stream, one more beach. An infinite and necessary being belongs to a categorically different class.

Conclusion

The ontological argument is not a proof in the strict logical sense that everyone will accept. But it points toward something genuinely interesting: the concept of an infinite, necessarily existing God has a special logical status that differs from ordinary empirical concepts. Just as infinity in mathematics is logically coherent and necessary rather than contingent, God’s infinity may be argued to imply necessary existence. The argument works best when understood as a demonstration that if the infinite is possible, it is necessary, and that the concept of God, when analyzed carefully, is precisely such a concept. Whether this is convincing depends, in the end, on whether one accepts that conceptual necessities can grip reality in the same way mathematics does.

References

• Anselm of Canterbury, Proslogion (1077-78).
• Descartes, R., Meditations on First Philosophy (1641).
• Godel, K., “Ontological Proof” (1970; published 1987, edited by Dana Scott).
• Pryor, J., “Guidelines on Writing a Philosophy Paper”.