That there should be a connection between theology and mathematics might seem strange to us. However, it was not a strange concept to many of the early modern scientists, men like Copernicus, Kepler, and Newton. In fact, according to Nancy R. Pearcey and Charles B. Thaxton:
...many of the early scientists were committed to a religiously grounded understanding of nature that had two major corollaries, one metaphysical and the other epistemological--that God made the world with a mathematical structure and that He made human beings with a mind capable of understanding that structure. (The Soul of Science: Christian Faith and Natural Philosophy, p. 130)
However, by the end of the 18th century, the connection between theology and mathematics began to break down. Most scientists had come to believe "that the universe was a perfectly running perpetual-motion machine--a view that eliminated the need for God to do anything except perhaps start it all off" (Pearcey and Thaxton, p. 137). Moreover, it was widely believed that '"the axiomatic method [as employed in mathematics] led to universal and absolute truth--a view that eliminated the need for divine revelation" (p. 137).
Thus, mathematics was essentially divorced from God. However, once mathematics was no longer grounded in the existence of a Creator, doubts about the certainty of mathematics began to arise. These doubts began when the Scottish philosopher David Hume (1711-1776) attacked the axiomatic method, which consists of a set of self-evident principles--axioms--and logical deductions from those principles. As Pearcey and Thaxton point out: "the use of the axiomatic method in science rested on the assumption that relationships in nature are necessary--that is, that the necessary connections traced by deductive reasoning have analogues in inexorable and immutable laws of nature" (p. 138). However, Hume argued, we have no way of knowing for certain that the deductive reasoning utilized by our minds actually has an analogue in the world existing outside our minds. We believe that fire causes heat because we normally experience the two phenomena together. This connection is created by our minds, but we have no way of knowing whether the connection actually exists in the world outside our minds. Thus, since there is "no physical necessity between events, there are no necessary laws of nature. Therefore, nature cannot be known by the axiomatic method, which traces necessary relations" (p. 138) and mathematics "does not give us knowledge of the physical world" (p. 138).
The German philosopher Immanuel Kant (1724-1804) was disturbed by Hume's claims. On the one hand, Isaac Newton's work had "provided a stunning practical confirmation that mathematics is an effective tool for gaining knowledge of the physical world" (Pearcey and Thaxton, p. 139). On the other hand, Kant realized that Hume was correct in criticizing the "standard arguments explaining why mathematics works" (p. 139). According to Kant, Hume was right in asserting that we have no reason for believing that the ideas in our minds correspond to the world outside our minds. Kant's solution to the problem was "to postulate that the mind itself creates order--which we then mistakenly regard as existing in the external world" (p. 139). In other words, Kant defined "order and design as a creation not of God but of the human mind--a product of its inherent structure." Unfortunately for Kant, mathematicians then discovered that "the human mind can conceive many different designs, all equally rational, all equally consistent" (p. 158). As a result, "mathematicians gave up the idea of truth and contented themselves with proving consistency [within mathematics]" (p. 139).
However, the effort to prove that mathematics was internally consistent foundered on the discovery by mathematician Kurt Godel (1906-1978) that formal systems like mathematics are "not capable of proving all true statements within the system" (p. 151). As Pearcey and Thaxton explain:
In order to prove that a formal system...is consistent, one must first be able to say definitely of any given formula that it is or is not provable within the system. Godel demonstrated that within any system it is possible to find the logical equivalent of a sentence that states, "This sentence is not provable"--which is self-contradictory. For if the sentence is true, then it means that the system contains at least one sentence (namely, this one) that cannot be proved within the system; but if the sentence cannot be proved, we don't know whether it is really true. It's a vicious cycle. (p. 150).
Godel's argument is known as the "incompleteness theorem." As Peter Biles at Mind Matters explains: "Godel thought that every mathematical system could not stand on its own without reference to something outside itself. Internal mathematical systems rely on truth outside the systems." In other words, as Biles explains:
Mathematics can't exist in a closed system, and is not merely an illusion, but "participates" in objective truths, according to Godel. In addition, because order, design, and meaning are apparent in the natural world, we can anticipate that material reality is not final.
Furthermore--I might add--if "material reality is not final," then there must be something or someone that exists beyond, or if you prefer, behind material reality. And it is not unreasonable to assert that what exists beyond or behind material reality is its Creator.
Thus, we come full circle. The earliest scientists believed mathematics reflected the mind of the Creator. Later, scientists (and mathematicians) came to believe that the certainty of mathematics was not dependent on God. However, Godel helped show that such a belief was unwarranted. Now, that is some food for thought!
Image of Kurt Godel from Wikimedia Commons