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Mathematics Can Prove the Existence of God

Atheist biologist Jerry Coyne finds that difficult to believe but it’s really a matter of logic

This story was #3 in 2022 at Mind Matters News in terms of reader numbers. As we approach the New Year, we are rerunning the top ten stories of 2022, based on reader interest. In “Mathematics can prove the existence of God” (July 31, 2022), neurosurgeon Michael Egnor offers this thought: Because mathematics can show infinity, eternity, and omnipotence, it can only have proceeded from a mind with those characteristics. That’s God.

In a recent post, atheist biologist Jerry Coyne takes issue with a commenter who asserts that God exists in the same sort of way mathematics exists. Here’s the analogy the commenter offered, as quoted by Coyne:

Think of numbers for example, or mathematical equations, these are metaphysical things, that have not been created, however were discovered. The number 7 was the number 7 before anything at all came into existence. This is also true concerning the nature of God. He is not some material being that has come into existence, he is like a number that has always existed, (and by the way nobody will deny this logic with the number, however when someone mentions God a problem occurs).

Jerry Coyne, “A new conception of God: He’s real in the way mathematics is” at Why Evolution Is True (July 20, 2022)

The commenter did not intend to prove God’s existence using mathematics. He merely pointed out that God’s existence is analogous, in limited ways, to the existence of numbers — they, like God, are immaterial, real, and eternal. Which, of course, is true. And Coyne will have none of it.

There is, in fact, a classical proof of God’s existence that uses universal concepts such as mathematics, proposed most prominently by St. Augustine (354–430 CE) of Hippo in the 4th century AD. It’s sometime called the Augustinian Proof*. I find it quite compelling and it goes like this:

Two kinds of things exist in the natural world: particulars and universals. Particulars are specific material things we know by our senses — a rock, a tree, my neighbor Joe, etc. Universals are abstract concepts that we know in the sense that we can contemplate them and talk about them — geology, botany, humanity, etc. But we cannot know any of these abstractions by our senses alone. We know abstractions by our intellect, which is our capacity for abstract thought.

Mathematics is an archetype of universals — take for example, the set of natural numbers. It includes all counting numbers—1,2,3,4 and so on. There has been some debate among philosophers and mathematicians about the reality of numbers (i.e. do they exist in a separate Platonic realm, or only in the human mind, or do they have no existence at all — in other words, are they merely words?). This is a profound question, but the view that natural numbers (and other universals) do exist in reality in some fashion is very hard to deny.

For example, consider the formation of our solar system. It formed around one sun, not two or three or a million suns — and it formed before there was any human mind to count the suns. But it is surely just as true that our solar system had one sun a billion years ago as it is true now. So the number 1 really exists in some fashion independent of the human mind. The same could be said of any number. For example, we know the ratios of many physical constants of the universe that have existed since the Big Bang, and because these ratios are real (we can measure them) then the numbers the ratios represent are real.

So how could numbers exist in reality, independent of the human mind? Plato proposed a realm of Forms in which universals exist, and in which our concepts participate. There are notorious problems with Plato’s concept of the realm of Forms (philosopher Edward Feser has a good discussion of this). But is seems undeniable that universals (such as numbers) do really exist in some real sense.

The solution proposed by Augustine (and many other philosophers and theologians, most notably Gottfried Wilhelm Leibniz) is called scholastic realism. Scholastic realism posits that God’s Mind is the Platonic realm of Forms. Augustine proposed that universals such as numbers, mathematics in general, propositions, logic, necessities and possibilities exist in the Divine Intellect, which is infinite and eternal.

What’s remarkable about the reality of universals as proof for God’s existence is that it points in a simple and clear way to some of God’s attributes, such as infinity, eternity, and omnipotence. To see how, consider again the set of natural numbers, which is infinite. Therefore:

  • The Mind that contains them must itself be infinite.
  • Because the Mind in which natural numbers exists is infinite, it is also omnipotent. Limitations on power are finite and are inconsistent with an infinite Mind.
  • Because numbers exist independently of the material universe, they are eternal (e.g., the truth that 1+1=2 is independent of time) and thus the Mind that contains them is eternal.

I find the Augustinian Proof of God’s existence via the reality of universals in the Divine Mind a compelling proof. It is a highly satisfying and an even beautiful concept — our abstract thoughts have a real existence in the Mind of our Creator, and we, who are created in His image, participate in His thoughts.

So while the analogy drawn by the commenter on Coyne’s blog between God and mathematics does not in itself demonstrate God’s existence (as the commenter acknowledges), the very existence of mathematics and other abstract concepts points in a quite direct way to an infinite, omnipotent, and eternal Divine Intellect.

Thus mathematics itself is a proof of God’s existence. As I have observed elsewhere, God is everywhere, if you know how to look for Him.

Note: *Philosopher Edward Feser has a very nice discussion of this proof in his superb book Five Proofs of the Existence of God (2017). I highly recommend it.

You may also wish to read: The Divine Hiddenness argument against God’s existence = nonsense. God in Himself is immeasurably greater than we are, and He transcends all human knowledge. A God with whom we do not struggle — who is not in some substantial and painful way hidden to us — is not God but is a mere figment of our imagination. (Michael Egnor)

Michael Egnor

Senior Fellow, Center for Natural & Artificial Intelligence
Michael R. Egnor, MD, is a Professor of Neurosurgery and Pediatrics at State University of New York, Stony Brook, has served as the Director of Pediatric Neurosurgery, and award-winning brain surgeon. He was named one of New York’s best doctors by the New York Magazine in 2005. He received his medical education at Columbia University College of Physicians and Surgeons and completed his residency at Jackson Memorial Hospital. His research on hydrocephalus has been published in journals including Journal of Neurosurgery, Pediatrics, and Cerebrospinal Fluid Research. He is on the Scientific Advisory Board of the Hydrocephalus Association in the United States and has lectured extensively throughout the United States and Europe.

Mathematics Can Prove the Existence of God