The God that is described in Abrahamic religions (such as Judaism, Christianity, and Islam), can be approached from a mathematical lens to reveal new insights. The main way I will do this is by using Gödel's Incompleteness Theorem to show how perfect omniscience is impossible. I will also explain the possibility of divine consciousness through Strange Loops, and describe God as Laplace's Demon, terms which will be explained later.
To understand divine omniscience mathematically, I must first establish what scripture claims about God's knowledge. According to Abrahamic tradition, God is all knowing. Psalm 139:1-4 states:
O Lord. You have examined me and know me. When I sit down or stand up You know it; You discern my thoughts from afar. You observe my walking and reclining, and are familiar with all my ways. There is not a word on my tongue but that You, O Lord, know it well.1
Mathematically speaking, there are many interpretations of what God's knowledge consists of, which I will describe as a single arbitrary set . But what might be in this set?
Pierre-Simon Laplace describes in his paper "A Philosophical Essay on Probabilities" a hypothetical being (which readers have come to call Laplace's Demon) that has the knowledge of the position and forces of every particle in existence.2 Let be the set of all knowledge which Laplace's Demon holds. Since the Abrahamic God is described in scripture as omniscient, this necessarily includes knowledge of the physical universe. We can therefore say that Laplace's Demon's knowledge is part of the Abrahamic God's knowledge, which can be denoted as (that all elements in set are also in set ). This means that God can calculate essentially any past and future state of the universe, assuming that physics is deterministic. But Laplace's Demon only knows the external universe: particles, forces, physical states. The Abrahamic God presumably also knows himself. This is where things get mathematically interesting. If God's omniscience includes self-knowledge, then must somehow contain information about itself. This self-referential quality is useful for understanding both the possibilities and limitations of divine omniscience.
The Abrahamic God can also think, which is evident throughout the Old Testament, New Testament, and Quran. For example, Genesis 18:17 describes God deliberating: "Now the LORD had said, 'Shall I hide from Abraham what I am about to do.'"3 Even a process as complicated as thinking can be mapped to mathematical terms. We have computers, which already loosely resemble thinking. We also have neural networks, which loosely resemble brains. Specifically, LLMs are even more closely related to language processing in humans, but this is not enough to accurately understand thinking in a mathematical sense since these technologies are commonly understood to not be directly mappable to biological processes. Instead, we can opt for what Douglas Hofstadter calls Strange Loops.4
But before we can understand what that is, we need to understand Formal Systems. A Formal System is a set of symbols, rules for manipulating those symbols, and starting assumptions called axioms. Think of it like a game: you have pieces (symbols), rules for how to move them (rules of inference), and a starting position (axioms). From these, you can derive new positions (theorems). Mathematics itself is a Formal System, where we start with basic axioms and derive new truths using logical rules.
Let's use the set created earlier for example. To reiterate, is the set of all particles, their positions, and their forces. The rules that we can apply to them are just the laws of physics. From any given state, we can derive past or future states. Now Strange Loops are a special kind of structure that can arise within Formal Systems, specifically, structures that contain self-reference. With humans in particular, we have self-reference in our minds as well. We have a representation of ourselves, for example the fact that we like to cook, or the sound of our voice, or that our body belongs to us. We aren't literally inside our own minds as an object, but our minds contain a representation of ourselves. This representation is why we have the concept of I.
This applies to humans, but it can also apply to the Abrahamic God. Firstly, Genesis 1:27 says "And God created man in His image," and therefore God may think in a similar way to us.5 Secondly, there's evidence for God's self-reference, for example in Genesis 6:7 "when the LORD said, 'I will blot out from the earth the men whom I created.'"6 The use of I demonstrates self-reference. So even if he doesn't think similarly to us, there's still evidence that he engages in self-reference, which is the key characteristic of a Strange Loop. According to Gödel, any sufficiently powerful Formal System capable of self-reference implies two limitations.
First, according to Gödel's Incompleteness Theorem, any sufficiently powerful Formal System capable of self-reference cannot prove all the statements within that system that are actually true.7 If the Abrahamic God thinks within such a Formal System, then his system could never prove all true statements within it. This becomes a substantial limitation for the moral rules that are set in Abrahamic scripture. Since, theologically speaking, the scripture is written by God himself, there may be true moral statements (such as whether a specific action is ultimately good or evil) that God cannot formally demonstrate to be true using the rules of his own system. This is a limitation of God's perfect omniscience. In other words, while God may have complete knowledge of the physical universe (like Laplace's Demon), his self-knowledge is limited by inherent mathematical constraints.
Second, no sufficiently powerful Formal System can prove that it won't produce a contradiction. This is the second part of the Incompleteness Theorem.8 If the Abrahamic God thinks within such a system, it means he cannot prove his own consistency. Arguably, Genesis contains evidence supporting this limitation. Genesis 6:6 says: "And the LORD regretted that He had made man on Earth."9 This suggests God was unable to foresee the outcome of his creation, a limitation on his omniscience. Additionally, some readers might see a tension starting in Genesis 18:26, "the LORD answered, 'If I find within the city of Sodom fifty innocent ones, I will forgive the whole place for their sake.'"10 Yet in Genesis 19:25, "He annihilated those cities and the entire Plain, and all the inhabitants of the cities and the vegetation of the ground."11 The text never confirms whether innocent people were or were not found; it simply proceeds to the destruction. Since all peoples have children, young children most likely lived in these cities, and they would not yet have had the chance to commit wrongdoing, meaning innocents would have been destroyed despite God's promise. These pieces of evidence of God's regret and apparent tension are consistent with what the Incompleteness Theorem predicts.
This may seem grim to Jews, Christians, and Muslims alike, as it challenges traditional conceptions of unlimited divine knowledge. If they accept this mathematical approach, which results in the Abrahamic God's imperfect omniscience, there's still good news. Since God is a Strange Loop, it means that he is conscious. According to Hofstadter, Strange Loops, which consist of self-reference and operations on that self-reference are the very basis of consciousness, or at least self-consciousness. According to him, there are many parts of the umbrella term consciousness, self-consciousness being one of them. This is what I referred to earlier as the concept of I. The same self-reference that limits God's provable knowledge is what enables God to have a genuine self.
There are, however, limitations to this approach. The greatest one being the possibility that divine knowledge transcends Formal Systems. Although in scripture there is evidence for human-like thought processes, he may think in a way that does not align with Hofstadter's consciousness theory. In addition, Hofstadter's theory assumes that consciousness emerges from physical and computational processes which operate as Formal Systems, but God may be beyond that. Exploring such possibilities lies beyond the scope of this article.
Additionally, the biblical evidence for the Abrahamic God's imperfect omniscience could be interpreted as being anthropomorphic. This could mean two things. Firstly, it could mean that the bible never literally claims in the first place that God has perfect omniscience. Instead, the bible was being hyperbolic, and intended to describe God as knowledgeable rather than literally omniscient. Secondly, it could mean that the verses which support God's imperfect omniscience were metaphors. Therefore, God never actually experiences regret or misunderstands occurrences in the scripture. Instead, the language used was actually describing something else.
Another possibility is that Strange Loops may not need to adhere to the Incompleteness Theorem. Although Formal Systems require Incompleteness, there is a chance that not all Strange Loops exist within a standard Formal System. The Abrahamic God may be one of those exceptional Strange Loops.
A follower of Judaism, Christianity, or Islam may also have qualms with this mathematical approach because it conflicts with orthodox theology. The common interpretation of scripture is that the Abrahamic God gave humans free will, which is supported by verses such as Deuteronomy 30:19:
I call heaven and earth to witness against you this day: I have put before you life and death, blessing and curse. Choose life-if you and your offspring would live-by loving the LORD your God, heeding His commands, and holding fast to Him.12
Therefore, God can't predict what humans can do because they fall outside of God's Formal System and act on their own accord and do not adhere to a deterministic world. Meanwhile this mathematical approach does assume deterministic processes for all matter including humans. Since there are many other opposing theories as to how the world operates, this mathematical approach may look different under these different theories.
With these limitations in mind, the God of Abrahamic religions, which is commonly seen as unscientific, may still be approached using math. Applying math in this context can reveal interesting insights like how Gödel's Incompleteness Theorem makes perfect omniscience impossible, or how divine consciousness is enabled rather than constrained through Strange Loops. However, the approach is limited by the assumption that the world operates deterministically, and that thinking adheres to Formal Systems.
Notes
1 Ps. 139:1-4 (New JPS Translation).
2 Pierre-Simon Laplace, A Philosophical Essay on Probabilities, trans. Frederick Wilson Truscott and Frederick Lincoln Emory (New York: Dover Publications, 1951), 4.
3 Gen. 18:17 (NJPS).
4 Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (New York: Basic Books, 1979), 10-15, 684-719.
5 Gen. 1:27 (NJPS).
6 Gen. 6:7 (NJPS).
7 Kurt Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I," in Kurt Gödel: Collected Works, Volume I: Publications 1929-1936, ed. Solomon Feferman et al. (Oxford: Oxford University Press, 1986), 144-195.
8 Gödel, "On Formally Undecidable Propositions," 144-195.
9 Gen. 6:6 (NJPS).
10 Gen. 18:26 (NJPS).
11 Gen. 19:25 (NJPS).
12 Deut. 30:19 (NJPS).