The God of Data Science: On Computational Metaphysics

The famous Austrian mathematician Kurt G?del (1906 – 1978), a giant of contemporary philosophical and scientific thought, was a strong believer in God. If we apply a political metaphor - for what is worth - to philosophy and science, according to their distance from theology, on the left we can find materialism and positivism, while on the right we can find idealism and metaphysics. G?del was definitely on the far right of the spectrum. Like Karl Popper, he did not agree with the ideas of the Vienna Circle. Both thinkers in fact rejected the Circle’s positivism, even if for different reasons.

G?del was far from the zeitgeist of the XX century. He thought he was free to believe in anything which was not logically contradictory. He declared himself to be a Christian, and that he believed in the hereafter and in the immortality of the individual soul. He also had an interest in computer science. Even if he maintained that the mind could not be reduced to the brain, he had a mechanistic idea of the functioning of the latter, similar to that of Turing, insofar the brain works essentially as a computer. At the same time, he criticized Darwinism as an algorithmic mechanism, which for this reason he deemed too simplistic. He had no interests in social issues (unlike Einstein, for example) but politically he supported Eisenhower after WWII.

It is not necessarily clear what G?del meant by God. In the first place, God is not necessarily the God of religions, but could be a principium primum, a cause without a cause, which is necessary to avoid the paradox of a regressus ad infinitum: without a beginning there would be nothing now, because it would require an infinite time. This God is impersonal, akin that of deism, a common belief among the philosophers of the Enlightenment, who were in most cases agnostics or atheists, at least in the sense that they denied the God of theistic religions.

In such religions (for example Christianity, Islam and Judaism) God certainly is a principium primum, but not only that. He is like a father, or anyway a person, and has many other positive qualities beside that of being the creator of the Universe. Probably, the God of G?del had both characteristics, being therefore similar to the God of Abrahamic religions.

In order to support his belief, that is his hypothesis, G?del wrote The mathematical proof of the existence of God. In his attempt, he adopted an ontological approach, following Catholic monk and Scholastic philosopher Anselm of Aosta (1033 – 1109), who was probably the first, historically, to use it. Aosta, in northern Italy, was his birthplace, but since he served as the archbishop of Canterbury since 1093 till his death, he is often known as Anselm of Canterbury in the English-speaking world.

The ontological proof is based exclusively on logic and deduction: starting from some premises, the conclusion (God exists) is necessary. It does not depend on empirical evidence. Further, ontological means that it revolves around the idea of being. G?del’s demonstration resembles that of Leibniz, in that God is conceived as ens perfectissimus, and it goes like this, in a very simplified version for the non-logician:

1. All positive qualities are mutually compatible, therefore a being with all positive qualities can exist (possibility).
2. Existence is a positive quality.
3. God has all positive qualities (including, therefore, existence), and only those.
4. Therefore, God exists (necessity).

The logic holds, apparently. Still, it is not so clear what G?del meant by “positive” in the first premise, but he probably meant in moral and aesthetic terms. Also, the second premise looks rather arbitrary: Is existence a quality? The third point instead is simply a definition, while the fourth is the necessary conclusion.

If you want to have more details about G?del’s proof, you should definitely read it. It is just a page in itself, but in most cases (unless you are very familiar with formal logic) you will need a commented version. I gathered most of the information about G?del and his proof from the Italian edition [1], especially from the preface by Gabriele Lolli. One of the editors, Piergiorgio Odifreddi, is a self-proclaimed atheist and tries to dismantle the proof of G?del (something which may appear a case of confirmation bias). In his opinion, it is correct but not convincing, also because it is quite simple to demonstrate something assuming it almost as a starting hypothesis, making the demonstration quasi-tautological.

I do not know whether the authors of the paper on the Formalization, Mechanization and Automation of G?del's Proof of God's Existence [2] were atheists or religious, but they merely meant to verify that the proof of G?del was formally correct [3]. They adopted a computational approach, which may be of interest to data scientists and computers scientists alike, beside people with some interest in metaphysics. Even German news outlet the Spiegel announced the publication of the paper in 2013 in an online article [4].?G?del’s proof is difficult to formalize and verify because it requires a language with modal operators of possibility and necessity (as you can see from the deductive schema above). Nonetheless, the authors of the study appeared to succeed, proving the effectiveness of modern deductive software applied to classic logic.

References

1 G?del K, La prova matematica dell’esistenza di Dio, a cura di Gabriele Lolli e Piergiorgio Odifreddi, Bollati Boringhieri, 2006.

2 Benzmüller C and Woltzenlogel Paleo B. Formalization, Mechanization and Automation of G?del's Proof of God's Existence. ArXiv abs/1308.4526, 2013. Frontiers in Artificial Intelligence and Applications, Volume 263: ECAI 2014.

3 Benzmüller C and Woltzenlogel Paleo B. Automating G?del's ontological proof of God's existence with higher-order automated theorem provers. In Proceedings of the Twenty-first European Conference on Artificial Intelligence (ECAI'14). IOS Press, NLD, 93–98, 2014.

4 Von David Knight, Computer Scientists 'Prove' God Exists, Spiegel International, 23-10-2013.