The surprising way mathematics applies to and describes the natural world is evidence for the existence of God. Several examples are necessary to elucidate this proposition. Complex numbers (numbers that contain the imaginary number, i) were formulated by Gerolamo Cardano in 1545 as an ingenious and beautiful solution to abstract algebra.[1]These numbers were not suggested by sensory experience or empirical measurement; they had no usefulness in the physical realm. Cardano himself is said to have described them as “worthless.” However, complex numbers were eventually used to describe real-world vectors, such as forces, velocities, and accelerations. Hundreds of years later, complex and imaginary numbers proved vital in disciplines such as electrical engineering, signal processing, and quantum mechanics.[2]
Quantum mechanics provides an additional example of physically applicable mathematics developed without any sensory experience or empirical measurement. The time-dependent Schrödinger equation also happens to employ the imaginary number. Schrödinger based his equation on de Broglie’s postulate that all matter had wave properties, but it is uncertain if Schrödinger knew exactly what physical occurrence his math was describing. Physicist Richard Feynman remarked that the equation came entirely out of the mind of Schrödinger.[3] Even so, it has turned out to be one of the most accurate mathematical descriptors of physical reality. Einstein’s famous equation, E=mc^2, also represents a mental production that demonstrates the relationship between mass and energy. Derived without any empirical evidence, Einstein’s famous equation has subsequently been shown to empirically describe the physical world perfectly.
Complex numbers, the Schrödinger equation, and E=mc^2 are only three of the many examples of mathematical concepts that arise in our mind and subsequently result in accurately describing natural laws in the physical realm. Physics is dependent upon mathematical concepts of this type that have unexpected and surprisingly accurate connections to physical phenomena.[4] In addition, and perhaps even more surprising, is that mathematics developed to model one specific situation often has several other future applications. Maxwell’s equations are a good example. Maxwell published his work demonstrating that light is an electromagnetic phenomenon in 1873, but the equations were not verified empirically until the 1890s.[5] They are now used as a mathematical model for anything related to light, electricity, or magnetism, such as power generation, electric motors, circuitry, and wireless communication. After discovering radio waves, Henrich Hertz remarked that since we are getting much more out of them than was originally put into them, Maxwell’s equations must have an independent existence.[6] Another example of a mathematical concept appearing repeatedly in different aspects of nature are trigonometric functions. Derived initially to describe the positions of stars, sine and cosine are now used to study additional phenomena such as rotational and oscillating motions, the vibrations of atoms and molecules, and even the fluctuations present in economics.
The above examples are representative of the commonplace occurrence of something we invent in our minds (math) matching perfectly with a universal physical law, often having unexpected applications in multiple areas. Granted, some equations are developed to match observations directly. As demonstrated, however, much of the math used to describe the physical world is not motivated by empirical adequacy. Instead, it arises through theoretical deduction and is often initially investigated and chosen for reasons such as beauty, fruitfulness, depth, and simplicity rather than empirical applicability.[7] In this sense, mathematical concepts arise from the aesthetic impulse in humans with no concern for empirical adequacy.[8] It is surprising that math chosen in this manner would subsequently be an accurate description of the natural world. The fact that the universe unexpectedly conforms to these a priori mathematical entities and concepts merits an explanation.[9] Why do mathematical concepts make entirely unexpected connections? How can the manipulation of human-invented symbols according to human-invented rules reveal information about the physical world – even information about phenomena beyond our senses, such as quantum particles and radio waves?
One possible answer to these questions, common among mathematicians, is that mathematical concepts are part of an abstract domain (the Platonic realm) separated in both time and space from the physical world.[10] This view holds that mathematical entities exist as real abstract objects in a metaphysical realm with no causal connection to the physical world. It would be an amazing and fortunate coincidence if the physical world conformed to these abstract and acausal entities. Mathematicians commonly provide unhelpful reasons such as “it’s a mystery,” or “it’s an undeserved miracle” to explain why the physical world conforms to the Platonic realm of mathematical objects.[11]
Another common answer to why mathematics works to describe the physical world is that math only exists as a helpful fiction invented in our minds. We made up the math to describe the physical world, so of course it describes the physical world! However, the above examples demonstrate that quite often the math is invented prior to seeing the relevance it has to the physical world. This view cannot explain why mathematical fiction, invented years before, has application to and is so effective in describing the physical world. Given these two choices (Platonism or fictionalism), it would be a fantastic coincidence if mathematical concepts were significantly effective in describing the physical world.[12] Neither of these two options adequately explain how a mathematical concept with no causal ability can, either in the Platonic realm or just in our mind, so accurately describe the physical universe.
Theism provides a more reasonable explanation of the fact that the laws of nature can be formulated as mathematical descriptions that are often significantly effective in physics. Many of the fathers of modern science, such as Isaac Newton and Johannes Kepler, viewed God as the explanation for why we could understand the world in the language of mathematics.[13] Kepler envisioned a three-part cosmic harmony. A mathematical rational plan (a blueprint) for creation has existed from eternity in the mind of God, a material manifestation of this plan is what we investigate, and the image of God enables humanity to study nature using observation and mathematics. Because humanity was created with the mind of God, we can have some access to God’s mathematical blueprint. The mathematics we discover applies to the physical world because it is the blueprint God used to create the universe and forms the basis of the physical laws established by God. Attempts to use Platonism or fictionalism as the explanation for the effectiveness of mathematics leaves it as a mystery or a fortunate coincidence. The physical applicability of mathematics is best explained by an intelligent creator and designer of the universe and provides evidence for the existence of God.
[1] James Nickel, Mathematics: Is God Silent? (Vallecito, CA: Ross House Books, 2001), 275-276.
[2] Sarah Salviander, “The Miracle of Math,” Schrödinger’s Poodle Blog (March 14, 2025). https://sarahsalviander.substack.com/p/the-miracle-of-math.
[3] Jürgen Renn, “Schrödinger and the Genesis of Wave Mechanics,” Max Planck Institute for the History of Science, Preprint 437 (2013), https://www.mpiwg-berlin.mpg.de/Preprints/P437.pdf.
[4] Eugene Wigner, “The Unreasonable Effectiveness of Mathematics In The Natural Sciences,” Communications in Pure and Applied Mathematics, Vol.13, No.1 (February 1960).
[5] Nickel, Mathematics, 219.
[6] Ibid., 220.
[7] Wigner, “The Unreasonable Effectiveness of Mathematics In The Natural Sciences.”
[8] Ibid.
[9] Dr. William Lane Craig, “The Argument From the Applicability of Mathematics,” Lecture given to CSSR 660, Summer Session, Biola University, June 10, 2025.
[10] Dr. William Lane Craig, “God and the Unreasonable Effectiveness of Mathematics,” Weekly Q and A with Dr. William Lane Craig, (January 11, 2019).
[11] Wigner, “The Unreasonable Effectiveness of Mathematics In The Natural Sciences.”
[12] Wigner, “The Unreasonable Effectiveness of Mathematics In The Natural Sciences.”
[13] Melissa Cain Travis, Thinking God’s Thoughts: Johannes Kepler and the Miracle of Cosmic Comprehensibility, (Moscow, ID: Roman Roads Press, 2022).
