“G‑d is a mathematician”
Carl Friedrich Gauss

I. Can We Prove that G-d created Axioms of mathematics?

1. Introduction

A reader challenged me with a question, “Could you prove G-d created basic propositions (axioms) of mathematics?” It is a profound question that merits a more detailed answer.

There is no universally accepted “proof” in the mathematical sense that G‑d authored the axioms (or “basic propositions”) of mathematics. The question of whether G‑d is the ground—that is, the metaphysical foundation—or source of mathematical truths is a longstanding philosophical and theological debate. Here are a few perspectives on this issue, along with reasons why no formal, universally agreed-upon proof exists.

2. What Would a “Proof” Even Look Like?

i. Nature of Mathematical Proof

In mathematics, proofs demonstrate that a conclusion follows logically from a set of axioms. Axioms themselves are taken to be true without proof. For instance, in Euclid’s Elements (c. 300 BCE), Euclid builds geometric theorems from his famous five postulates (axioms). However, Elements simply postulates (that is, assumes) those axioms are true and self-evident, rather than proving them in any sense. (Euclid, 1908/ original work c. 300 BCE) While axioms are no longer required to be self-evident, axioms are always necessarily taken (i.e., assumed) to be true at face value. The whole enterprise of axiomatic mathematics has the following form: assuming premises A, B, …, and C (axioms), what theorems can be logically derived from them?

However, if one aims to show that “G‑d is the source of these axioms,” that is no longer a mathematical statement. It becomes a metaphysical or theological claim. Mathematical systems, by design, do not attempt to prove or disprove axioms or prove the existence or acts of a divine being, which is the domain of theology.

ii. Philosophical vs. Theological Framework

Proving G‑d as the source of mathematical truths pushes the question from “What are our axioms?” to “Why should these axioms exist at all, and from where do they arise?” (Kant, 1787)

One must adopt specific philosophical or theological premises to talk about “proof” that G‑d created axioms. Different philosophical traditions (theistic, atheistic, deistic, etc.) each have different foundational assumptions. Any “proof” relies upon premises that may not be acceptable to all.

3. Traditional Lines of Argument

Despite the absence of a formal proof, there are classic arguments—some drawn from theology, some from the philosophy of mathematics—that try to connect G‑d with the existence of mathematical truths.

i. Platonist or Realist Views

Mathematical Platonism

Plato held that mathematical objects (such as numbers and shapes) exist in an eternal, unchanging realm of “Forms.” (Plato, 1925/ 4th century BCE)

Philosophers of mathematics expressed views on the nature of mathematical objects that fall into four categories: (a) Platonists, who believe that mathematical objects exist outside of spatiotemporal constraints and do not causally interact with physical matter; (b) physicalists, who believe that a number is a general category that describes the count of similar physical objects; (c) mentalists, who believe that numbers and other mathematical objects exist only in our imagination; and (d) counter-realists, who believe that mathematical objects do not exist. Among these approaches, the last three are often critiqued as incoherent, leaving Platonism as the only viable option.

Some mathematicians and philosophers have held that mathematical objects and truths exist independently of human minds in an abstract realm. (Gödel, 1947), (Penrose, 2004)

Some theologians maintain that if these eternal entities exist, G‑d could be responsible for sustaining that realm. (Plantinga, 1982)

G‑d as the “Divine Mind”

In this view, mathematical truths exist eternally in the mind of G‑d. (Augustine of Hippo, 395 CE/1993) The consistency and eternality of mathematics reflect the immutable nature of its divine source.

ii. Presuppositional Apologetics

G‑d as the Foundation of Logic and Reason:

Cornelius Van Til argued that the laws of logic and rational thought presuppose a theistic worldview. (Van Til, 1955) Others, like Greg L. Bahnsen, extended this argument to mathematics, suggesting that without G‑d as the ultimate ground, the consistency of logic and math lacks a transcendental basis. (Bahnsen, 1998)

Here, the argument is that logic, reason, and mathematics presume the order and rationality of the universe, which are said to be underpinned by the rational character of G‑d. Without G‑d—so this argument says—the axioms lose their transcendental grounding.

It is important to note that in Jewish theological tradition, we do not assert the rationality of G‑d, who is believed to transcend rationality. Of course, G‑d is the ultimate cause of the order and rationality in our world. However, as an unlimited being, He cannot be limited in any way. We cannot put G‑d in a straitjacket of rationality. This is expressed by the dictum nimna ha-nimna’ot (literally “restricting [all] restrictions”). (Rashba, 1997/ original work 1470), (Maharal, 1582), (Tzemach Tzedek, 1912) G‑d’s essence is utterly unknown to us and transcends any human notions of rationality and logic.

iii. Cosmological or Design Arguments

“Unreasonable Effectiveness of Mathematics”

Physicist Eugene Wigner famously noted that mathematics is “unreasonably effective” in describing physical reality. (Wigner, 1960) Some theists claim that G‑d’s creative design explains why mathematical structures map so well onto the physical universe. (Polkinghorne, 1991), (Dembski, 1999)

Mathematical Elegance and Natural Laws

For some, the elegance of fundamental equations in physics and other sciences suggests that a divine author is behind both the laws of nature and the mathematics describing those laws.

iv. Rational Intuition & Imago Dei

Another angle holds that humans can grasp necessary truths, including mathematical axioms, because humans are made in the image of G‑d, imago Dei (Genesis 1:26–27).

Our ability to “see” that certain propositions (like 1+1=2) must be true might reflect a divine spark or imprint.

4. Challenges to These Arguments

i. Alternative Explanations

A naturalist or atheist can argue that mathematics arises from fundamental human intuitions (e.g., about quantity, logic, patterns), which evolved because they conferred survival advantages or reflect consistent structures we observe in nature. Thus, Mill argued that mathematics is ultimately empirical and founded on inductive reasoning from experience. (Mill, 1843) On this account, there is no need for a divine source.

This argument has a fatal flaw—the inability to connect certain mathematical theories or concepts with the physical world. For example, mathematics often includes the concept of infinity. There are an infinite number of natural numbers. We have sets with an infinite number of elements, etc. However, there is no truly infinite object within the physical universe. Moreover, Georg Cantor showed that there are many different types of infinity, one larger than another, and an infinite number of them, which is demonstrated through the concept of cardinalities. We cannot even conceive these various infinities in the physical world. (Hilbert, 1925/1983) For this reason, the naturalist argument fails.

ii. Multiplicity of Axiomatic Systems

There is no single monolithic set of axioms for all mathematics; mathematicians often explore alternative axiom systems (e.g., non-Euclidean geometry, different logics, set theories). (Lobachevsky, 1829–1830), Which of these systems would be “the ones G‑d created” if G‑d were behind them? The diversity of mathematically valid frameworks can raise questions about the notion of G‑d as the singular designer of a unique set of axioms.

This objection is naïve, to say the least. First, many alternative geometries find realization in the physical world. Thus, Lobachevsky geometry is a special case of Riemannian geometry used in general relativity to describe gravity. Second, G‑d considers all possible worlds. The fact that we inhabit one of them implies no limitation on the creative power of G‑d, who can think of any mathematical theory, including theories based on mutually exclusive axioms.

iii. Language Games

In “Remarks on the Foundations of Mathematics,” Wittgenstein challenges traditional views about mathematical certainty and objectivity. His argument can be summarized as follows. Mathematics is not about discovering pre-existing truths but is a collection of diverse language games with rules established by human practices. Mathematical necessity is not metaphysical but grammatical – mathematical propositions function as rules for using language rather than descriptions of reality. Mathematical proof does not discover truths but generates new techniques and concepts that extend our mathematical practices. The certainty of mathematics is based on agreement in how we follow rules within language communities, not on correspondence to abstract objects. Mathematical understanding is demonstrated by knowing how to use mathematical concepts in various contexts, not by grasping abstract entities. The work opposes Platonism and suggests mathematics is a human creation embedded in life forms rather than a discovery of transcendent truths. (Wittgenstein, 1956/1978)

One can argue, however, that Wittgenstein completely ignores the “unreasonable effectiveness” of mathematics, reducing it to language games.

iv. Gödel’s Incompleteness Theorems

Gödel’s theorems show that within sufficiently strong axiomatic systems, there are true statements that cannot be proved from those axioms. (Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, 1931) This highlights the limitations of any axiomatic framework in capturing every mathematical truth. Whether one sees this as consistent with a divine mystery or merely an inherent mathematical property depends on one’s philosophical stance.

That said, if anything, Gödel’s theorems hint at the fact that there is a higher truth than what is captured by formal deduction. The existence of true statements that cannot be proven from the axioms of the theory may point to an ultimate source and arbiter of truth, namely, G‑d. No wonder Gödel himself was a believer and contributed to theology by formalizing Leibniz’s version of the ontological argument.

5. Why a Final, Universal Proof Is Unlikely

Difference of Starting Assumptions

If you begin with the premise that G‑d exists and is the source of all truth, then it is natural to interpret mathematical truth as an outflow of divine reason. If, however, you start with no such premise, then mathematics simply “is.” However, what that “is” really amounts to, we do not know. Mathematics “exists” in an abstract world—Platonic heaven—transcending time, space, and causality. There is no physical analog to such an existence. At the very least, mathematics presents a serious challenge to a physicalist view of the world.

Category Boundaries

Proofs in mathematics have a precise definition grounded in logical deduction from axioms. The claim that a transcendent Being established those axioms moves the discussion into theology and metaphysics. Such a claim falls outside the scope of what can be proven with the tools of mathematics alone.

Rational vs. Non-Rational Factors

Conviction in a theological conclusion often involves faith or revelation in addition to rational argument. Even brilliant philosophers and theologians who accept G‑d’s existence do so through a combination of reasoning, tradition, and personal experience—not solely by any mathematical or logical proof.

6. Conclusion

No consensus “proof” exists or can, in principle, exist that G-d created the axioms of mathematics because the very nature of the claim is metaphysical, and proofs in the strict mathematical sense do not operate in that domain. Instead, different worldviews offer different explanations for why mathematics is so consistent and universal. Theistic perspectives see mathematics as flowing from G‑d’s nature or creative act; atheists see it as a human-constructed framework (albeit one that accurately captures real structures in the universe), without postulating a divine source.

Ultimately, whether you interpret mathematical truths as divinely grounded, purely abstract, or emergent from human cognition depends on your overarching philosophical or theological commitments. Mathematical practice itself cannot resolve that question, because it does not—and cannot—reach outside its own axiomatic framework to show who or what established those axioms in the first place.

II. The necessary truths argument:

That said, the question we have explored in Part I can be restated in a form that is far more interesting. In philosophy, mathematical truths are considered necessary truths—it could not be otherwise. Or, using modal logic in the tradition of Leibniz, we can say that necessary truths are true in all possible worlds. A fundamental metaphysical question is, what gives necessary truths their necessity? The claim that “only G‑d could ground such necessary truths” means that only a necessary being (G‑d) can provide an adequate metaphysical foundation for truths that could not possibly be otherwise.

The argument that only G‑d could ground necessary mathematical truths has a rich philosophical history.

1. The Nature of Mathematical Necessity

Mathematical truths seem to be necessary rather than contingent (they could not be otherwise). Some philosophers argue that only G‑d could ground such necessary truths.

Mathematical truths appear to have a unique type of necessity. When we say “2+2=4” or “the interior angles of a triangle sum to 180 degrees in Euclidean geometry,” these statements seem necessarily true—they could not possibly be otherwise. Unlike empirical facts (like “water boils at 100°C at sea level”), which are contingent on how our universe happens to be, mathematical truths appear to transcend physical reality and would remain true even in vastly different possible worlds.

2. The Philosophical Challenge

This necessity poses a profound philosophical challenge: What grounds or explains these necessary truths? As Leibniz articulated in “Discourse on Metaphysics,” if mathematical truths are eternal and necessary, they must have some kind of eternal and necessary foundation. (1686//1991)

3. Formalization of the Argument:

Here is a formal rendering of the argument:

1. Mathematical truths are necessarily true (they could not be otherwise)
2. Necessary truths require a necessary foundation or grounding
3. Contingent beings or systems (including human minds, social conventions, or physical reality) cannot adequately ground necessary truths
4. Therefore, necessary truths must be grounded in something that itself has necessary existence
5. Only G‑d has the attribute of necessary existence
6. Therefore, G‑d is the foundation of mathematical truth

Augustine first developed a version of this argument in “De Libero Arbitrio.” (395 CE/1993) He reasoned that mathematical truths are eternal and unchanging, yet they are not physical objects. Since they transcend human minds (we discover rather than invent them) and physical reality, they must exist in the mind of G‑d.

Descartes advanced this line of thinking in his “Fifth Meditation” (1641). (Descartes, 1641/1984) He argued that mathematical truths are “eternal truths” that G‑d freely created but made necessarily true. For Descartes, the necessity of mathematics reflects G‑d’s immutability—G‑d does not change His mind about what is true.

Leibniz further refined the argument. In his correspondence with Clarke (1715-1716), he argued that necessary truths, including mathematical ones, exist in what he called “G‑d’s understanding.” G‑d contemplates all possible worlds, and necessary truths are those that hold in all these possibilities. (1956)

4. Alvin Plantinga’s Contemporary Formulation

In “Does God Have a Nature?” (1980), Plantinga presents a sophisticated version of this argument. He suggests abstract objects like numbers and mathematical truths are best understood as divine thoughts. As he puts it, “numbers and sets are best thought of as properties of G‑d,” specifically as “divine thoughts or concepts.” Plantinga argues that if mathematical truths were independent of G‑d, they would constitute a reality outside G‑d’s creative activity, limiting divine sovereignty. Instead, he proposes that these necessary truths are grounded in G‑d’s necessary nature.

This view resolves several metaphysical puzzles:

1. The ontological status of mathematics: Mathematical objects exist as divine thoughts rather than as mysterious Platonic entities or mere human conventions.
2. The necessity of mathematics: Mathematical truths are necessary because they reflect G‑d’s necessary nature.
3. Mathematical knowledge: Our ability to discover mathematical truths is explained by our being created in G‑d’s image, with minds capable of grasping divine mathematical thoughts.
4. The applicability of mathematics: The effectiveness of mathematics in describing physical reality makes sense if both mathematics and physical reality share the same divine source.

5. Revised Formal Argument

In section 3 above, we listed six steps of the formal argument to prove that G‑d is the foundation of mathematical truth. I find the second step— Necessary truths require a necessary foundation or grounding—rather questionable. Why does a necessary truth require a foundation or grounding? I propose another form of this argument which avoids this questionable step:

1. G‑d is necessarily limitless being
2. G‑d is a necessary being (He exists necessarily in all possible worlds)
3. Mathematical truths are necessarily true (they exist in all possible worlds)
4. If mathematical truths were to exist outside of G‑d, it would limit G‑d’s omnipresence in all possible worlds, which would contradict G‑d’ limitlessness (premise 1)
5. Therefore, mathematical truths necessarily exist in G‑d
6. Therefore, G‑d is the foundation of mathematical truth

1. Conclusion

The claim that “only G‑d could ground such necessary truths” means that only a necessary being (G‑d) can provide an adequate metaphysical foundation for truths that could not possibly be otherwise. This argument suggests several specific meanings of “grounding”:

1. Ontological dependence: Mathematical truths depend on G‑d for their existence. They exist as divine thoughts or as aspects of G‑d’s nature, rather than as independent realities.

• Explanatory foundation: G‑d explains why mathematical truths are necessary rather than contingent. Without G‑d, there would be no explanation for why “2+2=4” must be true in all possible worlds.

• Source of necessity: Mathematical truths derive their necessity from G‑d’s necessary nature. Just as G‑d cannot fail to exist, these truths cannot fail to be true.

• Causal foundation: In some versions (particularly Descartes’), G‑d causes mathematical truths to be necessary through divine decree.

• Truthmaker: G‑d serves as the truthmaker for mathematical statements—the reality that makes mathematical propositions true.

Different philosophers emphasize different aspects of this grounding relationship. For Leibniz, mathematical truths exist in G‑d’s understanding. For Augustine, they exist in the divine mind. For Aquinas, they are aspects of divine wisdom. For Plantinga, they are divine thoughts or concepts.

We also provide a novel proof that G‑d is the ground or foundation of mathematical truth using modal logic, which is more resistant to criticism.

The argument suggests that without G‑d, mathematical truths would either be:

• Inexplicably floating Platonic entities
• Merely human conventions lacking genuine necessity
• Brute facts without explanation
• Non-existent

The grounding relation offers a metaphysical account of why mathematical truths have the peculiar features they do: necessity, eternality, immutability, and intelligibility. The fact that the necessity of mathematical truths can only be explained by appealing to G‑d as their author provides another reason to say that it is more rational to believe in G‑d than not.

References

Augustine of Hippo. (395 CE/1993). De Libero Arbitrio (On Free Choice of the Will) (Vols. Book II, chapters 8-15). (T. Williams, Trans.) Hackett Publishing Company.

Bahnsen, G. L. (1998). Van Til’s Apologetic: Readings & Analysis. Presbyterian and Reformed Publishing Co.

Dembski, W. A. (1999). Intelligent design: The bridge between science and theology. InterVarsity Press.

Descartes, R. (1641/1984). Meditations on first philosophy, Fifth Meditation. In The philosophical writings of Descartes (J. Cottingham, R. Stoothoff, & D. Murdoch, Trans., Vol. 2, pp. 44-49). Cambridge University Press.

Euclid. (1908/ original work c. 300 BCE). The Thirteen Books of Euclid’s Elements. (T. Heath, Trans.) Cambridge University Press.

Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173–198.

Gödel, K. (1947). What is Cantor’s continuum problem? The American Mathematical Monthly, 54(9), 515-525. doi:10.1080/00029890.1947.11990229

Hilbert, D. (1925/1983). On the infinite. In P. Benacerraf, & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (2 ed., pp. 183-201). Cambridge University Press.

Kant, I. (1787). Critique of Pure Reason (2nd ed. (B edition) ed.).

Leibniz, G. (1686//1991). Discourse on Metaphysics. In R. Ariew, & D. Garber (Eds.), Philosophical Sssays (Vols. Sections 2, 13, and 30, pp. 35-68). Hackett Publishing Company.

Lobachevsky, N. I. (1829–1830). New Principles of Geometry with Complete Theory of Parallels.

Maharal, T. (1582). Gevurot Hashem. Frankfurt am Main.

Mill, J. S. (1843). A System of Logic, Ratiocinative and Inductive.

Penrose, R. (2004). The road to reality: A complete guide to the laws of the universe. Jonathan Cape.

Plantinga, A. (1980). Does God have a nature? Marquette University Press.

Plantinga, A. (1982). How to be an anti-realist. Proceedings and Addresses of the American Philosophical Association, 56(1), 47-70. doi:10.2307/3131293

Plato. (1925/ 4th century BCE). Timaeus. In Plato in Twelve Volumes (Vol. 9). (W. Lamb, Trans.) Harvard University Press.

Polkinghorne, J. (1991). Reason and reality: The relationship between science and theology. Trinity Press International.

Rashba, T. (1997/ original work 1470). Responsa of the Rashba (Shu”t HaRashb”a) (Vols. Vol. I, sec. 418). (H. Dimitrovsky, Ed.) Jerusalem: The Mossad Harav Kook.

The Leibniz-Clarke correspondence: Together with extracts from Newton’s Principia and Opticks. (1956). In H. G. Alexander (Ed.). Manchester University Press.

Tzemach Tzedek. (1912). Sefer HaChakirah. Jerusalem.

Van Til, C. (1955). The Defense of the Faith. Presbyterian and Reformed Publishing Co.

Wigner, E. P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, vol. 13(1), 1–14.

Wittgenstein, L. (1956/1978). Remarks on the foundations of mathematics. In G. H. von Wright, R. Rhees, & G. E. Anscombe (Eds.). MIT Press.

Addendum

Below is a representation of our six-step argument using the idiom of (propositional) modal logic—in particular, a normal system at least as strong as S5 (which is often used for “metaphysical necessity”). Readers unfamiliar with modal logic symbolism may safely skip to the Conclusion.

1. Identify the Core Propositions

We have these English statements:

1. G‑d is a necessarily limitless being.
2. G‑d is a necessary being (He exists necessarily in all possible worlds).
3. Mathematical truths are necessarily true (they exist in all possible worlds).
4. If mathematical truths were to exist outside of G‑d, it would limit G‑d’s omnipresence in all possible worlds, contradicting G‑d’s limitlessness (premise 1).
5. Therefore, mathematical truths necessarily exist in G‑d.
6. Therefore, G‑d is the foundation of mathematical truth.

We want to express these claims as modal formulas. We will introduce propositional symbols and, where needed, use standard modal operators:

□ϕ reads, “ϕ is necessarily true.”

◊ϕ reads, “ϕ is possibly true.”

We also assume “G‑d exists” or “G‑d is present” in some modal sense—commonly expressed as □G
(“G‑d necessarily exists”) if G = “God exists.”

Notation Choices:

1. Let □L = “G‑d is necessarily limitless (omnipresent, not limited in any world).”
2. Let □EG = “G‑d is a necessary being” (i.e., G‑d exists in all possible worlds).
3. Let □M = “Mathematical truths are necessarily true” (they hold in every possible world).
4. Let In(M, G) = “Mathematical truths exist in G‑d.” (This captures “G‑d is the ground or locus of mathematical truths” or “Mathematics does not exist outside G‑d.”)

The crucial statement about limitlessness (premise 4) can be formalized to say: “If M were outside G‑d, that would imply ¬L.” In propositional logic, we might treat “M is outside God” as a separate proposition O. But, since we tie it to necessity/possibility, we prefer a conditional: “□M ∧ Outside(M,G) ⇒ ¬L”.

Bearing all this in mind, below is the formal version of my argument.

Argument formalized in modal logic

A streamlined symbolic presentation might look like this:

□L (“G‑d is necessarily limitless.”)

□EG (“G‑d necessarily exists.”

□M “Mathematical truths are necessarily true.”

(□M ∧ Outside(M,G))→¬L (“If mathematical truths existed outside G‑d in any world, that would contradict G‑d’s limitlessness.”

□ In(M,G) (“Therefore, mathematical truths necessarily exist in G‑d.”)

F (or “□ In(M,G) → F”) (“Hence, God is the foundation of mathematical truth.”)  

We can indicate that (5) follows by noting: from (4) and (1), we eliminate the possibility of “Outside(M,G),” so the only remaining option is “In(M,G)”—and with □M, that leads to “□In(M,G).” Then (6) is more of a theological or metaphysical interpretation of (5).

Notes on Validity vs. Soundness

Validity: The above is a valid argument in propositional modal logic (with normal inference rules). That is, if we accept each premise as true, then the conclusions follow logically.

Soundness: Whether the premises themselves are true is a separate philosophical or theological issue. For instance, a critic might challenge the idea of limitless G‑d on the grounds that such a definition might be incoherent. Nevertheless, formally, we have a consistent set of premises that entail the conclusion in standard modal logic.

1. Summary

A concise symbolic form, consistent with S5-style modal reasoning, could look like this:

1. □L (“G‑d is necessarily limitless”)
2. □EG(“G‑d necessarily exists”)
3. □M (“Mathematical truths hold,”)
4. (□M ∧ Outside(M,G))→¬L (“If mathematical truths were to exist outside G‑d, that would contradict 1”)
5. ∴□ In(M,G) (“Therefore, mathematical truths exist in G‑d,”)
6. ∴F. (“Therefore, G‑d is the foundation of mathematical truth.”)

Steps (5) and (6) follow from (1) – (4) by standard rules of modal propositional logic (modus tollens, necessity distribution, etc.), with any minor bridging premise that “□ In(M,G) implies G‑d is foundational for math.”   Keeping in mind that absolute G‑d is identical with His divine attributes, and, therefore, the “mind of G‑d” is identical with G‑d’s essence, we can see that, at the end of our version of the argument, we arrive at the same conclusion as Plantinga, but without the second premise of his argument that may be challenged.