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Apparently Kurt Gödel believed that his incompleteness theorems have some kind of religious implications. Despite Gödel's belief in a personal God, this was still somewhat surprising to me. Discussions and theories about weird (i.e. outside of mathematics) consequences of his theorems are all over the internet, and are often labeled as misunderstandings or "crank" interpretations of his work. But Gödel himself seemed to think that there are indeed legitimate applications of his work to religion.

I recall reading the quote below a while ago. My memory is a bit fuzzy, but I believe it was in response to Kurt Gödel having heard from his mother that a religious magazine or journal of some sort printed an article describing a simplified account of his incompleteness theorems for a general audience. The article then discussed some religious implications.

The actual quote from Gödel is:

It was something to be expected that sooner or later my proof will be made useful for religion, since that is doubtless also justified in a certain sense.

The quote can be viewed on page 125 of Reflections on Kurt Gödel by Hao Wang, on Google Books as a preview. The context I described above is not there in the preview exactly as I remember, so I'm pretty sure I read it somewhere else (or am going insane). I do not have a copy of Wang's book either, so if anyone else wants to provide additional context beyond the preview or from other sources that is great.

My question is: What religious implications did Kurt Gödel think his incompleteness theorems have, and why?

My question is mainly about Gödel's own thoughts, but if anyone wants to speculate or "connect the dots" based on any other information they might have about Gödel's writing or thinking on the matter, this is more than welcome too.

  • I'll also be fascinated by the answers. Even mathematicians seem to fall-out on this question. It occurs to me that the question might be little different but more useful if it was asked about metaphysics rather than religion. If incompleteness has implications for religion it is because it has them for metaphysics. – PeterJ Apr 3 at 12:14
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    The incompleteness theorem is about arithmetic and no more. Unless God is a number, the incompleteness theorem can have no religious consequences, and even what Godel himself has to say about it can't change this fact. Other theorems like Tarski's undefinability theorem demonstrate more generally how language can't express every truth. However, these are demonstrations of the limits of language and the human mind only, and show nothing about metaphysics or religion. – armand Apr 8 at 7:51
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Gödel's theism is discussed by Franzen in Gödel’s Theorem: An Incomplete Guideto Its Use and Abuse. He penned a version of the ontological argument, and in 1961 ranked the worldviews “according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion)... Skepticism, materialism, and positivism stand on one side; spiritualism, idealism, and theology on the other”. Idealism "in its pantheistic form” is dismissed as as “a weakened form of theology in the proper sense”. Nonetheless, he did not attempt to draw theistic conclusions from the incompleteness theorem:

"Gödel sometimes described himself as a theist and believed in the possibility of a “rational theology,” although he did not belong to any church. In [Wang 87] he is quoted as remarking that “I believe that there is much more reason in religion, though not in the churches, that one commonly believes...” Among his unpublished papers was a version of St. Anselm’s ontological proof of the existence of God. More precisely, the conclusion of the argument is that there is a God-like individual, where x is defined to be God-like if every essential property of x is positive and x has every positive property as an essential property. As this explanation of “God-like” should make clear, Godel’s idea of a rational theology was not of an evangelical character, and Oskar Morgenstern relates ([Dawson 97, p. 237]) that he hesitated to publish the proof “for fear that a belief in God might be ascribed to him, whereas, he said, it was undertaken as a purely logical investigation, to demonstrate that such a proof could be carried out on the basis of accepted principles of formal logic.” Although Gödel was thus not at all averse to theological reasoning, he did not attempt to draw any theological conclusions from the incompleteness theorem."

This did not stop others from doing just that, or even ascribing it to Gödel. Much of it is also discussed by Franzen: there can be no "theory of everything", existence of truths which can not be mechanically derived imply the existence of God, for ultimate truth is beyond reason, methodology of science cannot be based upon science only, scientists must rely on faith as much as non-scientists, finite beings can never answer all the questions they seek after, etc., etc. Related, although not exatly theological, is the Penrose-Lucas argument that "consciousness" surpasses Turing machines. For a recent sampler, see e.g. Goldman's God of Mathematicians:

"At twenty-five he ruined the positivist hope of making mathematics into a self-contained formal system with his incompleteness theorems, implying, as he noted, that machines never will be able to think, and computer algorithms never will replace intuition. To Gödel this implies that we cannot give a credible account of reality without God.

[...] Whether or not we believe, as did Gödel, in Leibniz’s God, we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.

Other attempted drawings of implications suffer from similar reasoning by loose association, they are not so much implications as vague analogies. And while it is not clear that Gödel's God was Leibniz's God exactly (as opposed to, say, Anselm's), it is true that Gödel was quite preoccupied with Leibniz himself, see Why did Gödel believe that there was a conspiracy to suppress Leibniz's works? He even told Hao Wang:"My theory is a monadology with a central monad [namely God]. It is like the monadology by Leibniz in its general structure". Unfortunately, Gödel's surviving writings on this theory, and theology generally, are very scarce. His notes on philosophy, known as Max Phil (Maximen Philosophie), occasionally touch on theological issues, Ternullo in Gödel’s Cantorianism discusses Gödel’s views of the "absolute infinite", which Cantor associated with God.

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    Thank you. Given that "he did not attempt to draw any theological conclusions" together with my quote of him above, it seems that Godel may have been open (and even welcome) to theological implications, but he never actually tried to work out any himself. Personally I think it's a shame that Godel didn't (as far as I could tell) ever publish a "mature" philosophy of religion. Most of what I could find (on the web anyways) about Godel's theistic worldview, besides his ontological argument, are cryptic fragments and informal letters to his mother about his thoughts on an afterlife. – Adam Sharpe Apr 1 at 22:49
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    Another thing. When you wrote: "And while it is not clear that Gödel's God was Leibniz's God exactly (as opposed to, say, Anselm's)", are you referring to the theistic personalist vs. the classical theist conception of God? – Adam Sharpe Apr 1 at 22:53
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    @Adam I am not sure we have enough information to decide what Gödel's God was. He did say to Wang “My theory is a monadology with a central monad [namely God]. It is like the monadology by Leibniz in its general structure”. His notes, named Max Phil, allude to "rational theology". His only continuous theological text is a reworking of Anselm's argument. Ternullo tries to extract something from his Platonism and affinity to Cantor's "absolute infinite". – Conifold Apr 1 at 23:43
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    Thanks @Conifold for an excellent conspectus of the situation. You do mention Gödel's relation with the ontological argument. Maybe good to also put a link? (such as en.m.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof ) – Rusi-packing-up Apr 2 at 3:36
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    @Rusi I added the link. – Conifold Apr 2 at 18:26

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