The Timpson Lyceum · History of Philosophy (2024)

Proving God (Anselm, Thomas Aquinas, and Kurt Godel’s Proofs)

Overview

This lecture surveys the major philosophical arguments for the existence of God, tracing a single tradition from the medieval a priori proof of Anselm of Canterbury (1033–1109), through the a posteriori Five Ways of Thomas Aquinas (1225–1274), to their modern descendants (the fine-tuning argument) and, finally, the fully formalized modal-logic proof of Kurt Gödel (1906–1978). The aim is expository — to make each argument intelligible — rather than to evaluate or refute it. The throughline is a steady increase in logical rigor: from Anselm's conceptual reductio, to Aquinas's causal reasoning grounded in Aristotelian metaphysics, to Gödel's axiom-driven derivation in second-order modal logic.

These arguments matter because they define the classic options in natural theology — attempts to establish God's existence by reason alone, without appeal to revelation. Anselm's ontological argument reasons purely from the concept of God; Aquinas's Ways reason from observed features of the world (motion, causation, contingency, gradation, order); and the modern design and fine-tuning arguments update the fifth Way with contemporary cosmology and biology. Gödel's proof, examined line by line in the following lecture, recasts Anselm's insight in symbolic form, making its assumptions explicit and its validity mechanically checkable.

Key Concepts

• Ontology — The branch of metaphysics studying being qua being — existence in and of itself. Anselm's and Gödel's proofs are "ontological" because they argue from the very concept of a supremely perfect being.
• Anselm's definition of God — "that than which nothing greater can be conceived" — the greatest conceivable being. The whole ontological argument turns on this definition.
• Ontological argument — An a priori reductio: if God existed only in the intellect, one could conceive a greater being (one that also exists in reality), contradicting the definition; therefore God must exist in reality (see the worked example).
• A priori vs. a posteriori — Anselm's argument is a priori (from concepts alone, prior to experience); Aquinas's Five Ways are a posteriori (from observed facts about the world).
• The First Way (argument from motion) — Things are in motion; whatever moves is moved by another; this chain cannot regress infinitely (an actually completed infinite series could never have been traversed to reach the present), so there must be an unmoved mover — God.
• The Second Way (argument from first cause) — Using Aristotle's efficient cause (the agent that brings a thing about, as a sculptor causes a statue). Everything caused is caused by another; nothing causes itself; the causal chain cannot regress infinitely; hence a first uncaused cause — God.
• The Third Way (argument from contingency) — A contingent being can exist or not (it comes to be and passes away); a necessary being cannot fail to exist. If everything were contingent, there would have been a time when nothing existed, and since nothing comes from nothing, nothing would exist now — contradicting present existence. So a necessary being exists — God.
• The Fourth Way (argument from degrees) — Things exhibit gradations of perfection (better/worse, more/less), and such comparisons presuppose a maximum standard. That maximum of being — the cause of being in all else — is God. (Aquinas argues within an Aristotelian framework in which the maximum causes the participation of all lesser instances.)
• The Fifth Way (argument from design / teleology) — Natural bodies reliably act toward ends (an acorn toward an oak, planets toward orderly motion); complex ends achieved without fail bespeak a guiding intelligence, hence a designer — God. Teleology (from telos, "end/purpose") names this goal-directedness.
• Paley's watchmaker — William Paley's analogy: stumbling on an intricately ordered pocket watch, one infers a watchmaker; the ordered universe likewise implies a designer.
• Fine-tuning argument — A modern design argument: the physical constants (gravitational, electromagnetic, strong and weak nuclear forces, the cosmological constant) and initial conditions of the universe are set so precisely that tiny changes would preclude any complex structure or life — suggesting a fine-tuner. It claims not impossibility of chance but its overwhelming improbability.
• Gödel's ontological proof (preview) — A rigorous formalization of Anselm's insight in modal logic (S5), built from a predicate $P$ for positive properties, five axioms, and definitions of God-like, essence, and necessary existence, deriving $\Box\,\exists x\, G(x)$ — that a God-like being necessarily exists.

Worked Example

Anselm's ontological argument as a formal reductio. Let $G$ name the individual God, and read $\exists!$ informally as "there is such a being." Define the property $\text{Greater}(a, b)$ as "$a$ is greater than $b$," and split existence into existing in the intellect ($\text{InInt}$) versus in reality ($\text{InReal}$). Anselm's key premise is that real existence adds greatness.

1. Definition. God is that than which nothing greater can be conceived:

$\neg\,\Diamond\,\exists x\,\big[\text{Greater}(x, G)\big].$

That is, it is not even conceivable that something be greater than God.

1. Premise (existence-in-the-mind). God exists at least in the intellect — the concept is understood even by "the fool": $\text{InInt}(G)$.
2. Premise (reality outranks intellect). For any conceivable being, existing in reality is greater than existing in the intellect alone:

$\forall x\,\big[\big(\text{InInt}(x)\wedge\neg\,\text{InReal}(x)\big)\;\rightarrow\;\Diamond\,\exists y\,\text{Greater}(y, x)\big].$

1. Assumption for contradiction. Suppose God exists only in the intellect: $\text{InInt}(G)\wedge\neg\,\text{InReal}(G)$.
2. Apply step 3 to $G$. Then it is conceivable that some being $y$ is greater than God — namely one exactly like God but also existing in reality:

$\Diamond\,\exists y\,\text{Greater}(y, G).$

1. Contradiction. Step 5 directly denies step 1 ($\neg\,\Diamond\,\exists x\,\text{Greater}(x, G)$). The assumption in step 4 is therefore false.
2. Conclusion. Negating step 4, and given $\text{InInt}(G)$ from step 2, God must exist in reality:

$\text{InReal}(G). \qquad \blacksquare$

The move under scrutiny is step 3 — the assumption that real existence makes a being greater — which is precisely where later critics (and Gödel's more careful formalization) concentrate their attention.

Summary

• Natural theology seeks to prove God by reason alone; the lecture surveys the classic arguments expositorily, from Anselm to Gödel, with rigor increasing across the tradition.
• Anselm's ontological argument reasons a priori from the definition of God as "that than which nothing greater can be conceived," using a reductio to conclude God exists in reality.
• Aquinas's Five Ways reason a posteriori from the world: the unmoved mover (motion), the first cause (efficient causation), a necessary being (contingency), a maximum standard (degrees), and a designer (teleology).
• The design argument lives on in the fine-tuning argument, which stresses the extraordinary precision of physical constants and initial conditions as evidence of a fine-tuner.
• Gödel's proof formalizes the ontological insight in modal logic (S5) from axioms about positive properties, deriving the necessary existence of a God-like being — the subject of the next lecture.