Can Mathematics Prove The Existence Of God?
By Emanuel Rutten
January 31, 2017
Two computer scientists say they have proved a theorem by mathematician Kurt Godel.
A recent Patheos article talks about two computer scientists who claim to have applied state-of-the-art software to show that Kurt Gödel's "mathematical proof" for the existence of God is entirely correct.
In 2013 the computer scientists C Benzmüller from the Free University of Berlin and BW Paleo from the TU Wien indeed demonstrated through the application of specialised software that Gödel's conclusion that God exists logically follows from his assumptions or premises. What both scientists have thus shown is that Gödel's conclusion is true if we take it that his premises are true.
Does this mean that we now have a successful proof of the existence of God? No, clearly not. The computer scientists have only shown that Gödel's derivation of his conclusion from his premises is valid. Gödel's conclusion can be properly deduced from his premises. Gödel's deduction is logically valid. But this should of course hardly surprise us. After all, Kurt Gödel is not just by coincidence seen as one of the greatest logicians since Aristotle.
Whoever accepts Gödel's premises has indeed no choice but to accept the conclusion that God exists. But should we accept Gödel's premises? That's the real question. How sure can we be of the truth of these premises? Benzmüller and Paleo remain totally silent about this. Of course they do. Gödel's premises are not of a logical or a mathematical nature. They thus cannot be established or verified by a computer. Gödel's premises are so called philosophical or more precisely metaphysical premises, that is to say, premises about the fundamental nature of reality. One premise holds that there are "positive" properties and his other premises state specific ways in which these properties are mutually related. A computer cannot tell us anything about whether these premises are true.
This would not be a problem provided that Gödel's premises are sufficiently self-evident. For in that case, we would have no choice but to accept them. Hence in that case we would have indeed a proof of God's existence. The problem though is that Gödel's premises are anything but self-evident. There is in fact a lively debate amongst metaphysicians of whether they can be justified. It is for example not fully clear how the "positive" properties these premises refer to are to be understood. At most these premises can be said to be plausible. But then the most we can say is that the conclusion entailed by these premises is plausible. Or even more modest: the premises merely increase the plausibility of the existence of God. And this comes obviously not even close to a proof of God's existence.
That Gödel's argument does not count as a proof of God's existence is hardly surprising. For the existence of God simply cannot be proven. The best we can do is to provide reasonable arguments for the existence of God. Such arguments show that it is reasonable to think that God exists. They make God's existence plausibly true. In fact, these arguments taken together show that theism is the most reasonable worldview. But absolute and infallible certainty? A proof of God's existence? No, that is something these arguments cannot provide. This is a good thing. Reasonable arguments for God's existence give us reasons to believe, but they remain reasons to believe. They don't amount to a proof, but they make believe in God entirely reasonable. Or as St. Anselm of Canterbury has so aptly expressed it: fides quaerens intellectum, "Faith in search of understanding".
Gödel's argument for God's existence falls within the category of ontological arguments. Ontological arguments depart from pure thought and make no reference to sensory experiences. Their start from a specific definition of God, such as Anselm's definition that God is "that, than which nothing greater can be conceived", or that God is a "maximally perfect being". From such a definition the existence of God is step by step deduced through strict logical reasoning by making use of certain premises.
The first ontological argument in history is from Anselm. His argument is only convincing in a strict neo-Platonic context. Gödel's argument requires no neo-Platonism. It is thus a quite different type of ontological argument from Anselm's. In recent years, Gödel's argument has been significantly improved by Alexander Pruss. He has proposed alternative premises that are more plausible.
As discussed, Gödel's argument is based on metaphysical assumptions about the general nature of reality. Gödel subsequently applies modal logic (ie, the logic of the possible and the necessary) to deduce the conclusion that God exists. Therefore, the argument is not a mathematical derivation. Of course not. No collection of mathematical theorems entails the conclusion that God exists. At best, it can be argued that we need a reasonable explanation for why the cosmos is so perfectly described though mathematics, and that the best explanation is that the cosmos was created by a rational, all-powerful creator who wanted us to rationally understand it. But this again is no proof of God.
In addition to ontological arguments, there are other types of arguments for the existence of God that make use of modal logic, such as the modal-epistemic argument for God's existence explained here. Together, these arguments make God's existence likely, especially if we add to this set the various cosmological, teleological, moral and aesthetic arguments for the existence of God. Nowadays we do actually have a strong cumulative rational case for God's existence. But infallible certainty? A proof of God's existence? No, obviously not. We provide proofs in mathematics and in logic, not in philosophy.
Finally, which God is being referred to by Gödel's argument? Is it the God of Christianity or the God of another monotheistic tradition? More precisely formulated, the conclusion of Gödel's argument has it that there exists a being that possesses all "positive" properties. This amounts to the existence of a necessarily existing and perfect being that is the ground and origin of reality as a whole. Now, if such a being exists, it can surely be called God. But it requires further work to argue that this God is in fact the God of Christianity. Gödel's argument does nothing to show that the God inferred is actually the God that Christianity has testified for during centuries. It's an argument for theism, not Christianity.
Dr Emanuel Rutten is a researcher and lecturer in philosophy at the Faculty of Humanities at Free University of Amsterdam
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