In this blog post, I will examine the relationship between science and religious belief, focusing on Gödel’s Incompleteness Theorems, and consider whether science can become independent of religious belief.
Throughout history, humanity has sought its own answers to natural phenomena, and this process has driven the development of science. In eras when understanding of nature was limited, natural phenomena were explained through religious belief; mythical explanations of volcanoes, lightning, and the movements of celestial bodies are prime examples. However, as our understanding of nature deepened, the idea that the path to truth lies not in religious belief but in human reason became the prevailing spirit of the age, and as a result, religious belief gradually gave way to logic and reason.
Within this spirit of the age, truth based on a complete logical system was an object of long-standing human yearning, and its existence and discovery held significance akin to the “Promised Land” prophesied by the prophets. In particular, from the time of Pythagoras through the late 19th century, the mainstream scientific movement regarded mathematics as a map leading to this “promised land.” This absolute trust in mathematics can also be glimpsed in ‘Logicomica’, specifically in the passage that describes mathematics as the “queen of the sciences.” This faith in mathematics reached its peak in Hilbert’s Program. In a lecture in 1900, he declared, “There is nothing in mathematics that we will never know,” a statement based on the belief that the scope of truth is infinite and that every true proposition in mathematics can ultimately be proven.
However, the belief that mathematics is a sure map leading to the Promised Land faced crises on several occasions. The discovery of imaginary numbers, the concept of infinity, the concept of infinitesimals—which was not strictly defined and is also mentioned in ‘Logicomix’—and Cantor’s conclusions regarding infinite sets are prime examples of these crises. Russell’s paradox, which is a central theme in ‘Logicomix’, was also one of the key factors that led to these crises.
Among the various events that brought crises to mathematics, it was Gödel’s Incompleteness Theorems that, above all, shattered the belief in the completeness of mathematics. Simply put, Gödel’s Incompleteness Theorems state that among true propositions, there exist some that cannot be proven. This is the exact opposite of Hilbert’s confident declaration discussed earlier. Gödel’s theorem revealed that mathematics cannot be built on a completely solid foundation from the outset, but rather finds itself in a precarious position, as if standing precariously atop an endlessly stacked tower of turtles.
Here, “completeness” refers to the property whereby every true proposition within a given logical system can be proven within that system.
Some have interpreted Gödel’s Incompleteness Theorem as suggesting that mathematics could be viewed as a branch of theology. This perspective is based on the fact that while certain propositions may be intuitively accepted as true, it has been shown that they cannot be proven. This is because, in order to develop the system of mathematics based on such propositions, one must assume that the propositions are true—even in the absence of proof. When we consider the implications of this incompleteness theorem, we realize just how insignificant humans are in the face of the arduous path to truth, despite Hilbert’s confident declaration discussed earlier. Humans, who had been advancing ceaselessly alongside the development of mathematics, have instead discovered their own limitations through that very progress.
So, does this mean that modern science—having lost its firm foundation for advancing toward truth due to the Incompleteness Theorems—must now begin anew on the basis of religious faith? Is the tower of science, which humanity had been confidently building toward truth, ultimately nothing more than another modern religion based on religious faith? Does the realm of God still exist in the gaps of truth that science cannot ultimately uncover? Although the Incompleteness Theorem shattered the belief that mathematics is a complete system leading to truth, it is the author’s view that science, in its pursuit of truth, can nevertheless break free from religious faith.
Unlike religion, science is based on experiments that anyone can attempt, repeatability, and observation. Of course, inductive inquiry—a key methodology of modern science—also begins with a certain premise, in that research is conducted based on hypotheses established by the researcher or the prevailing paradigm of the time. However, these premises are fundamentally different from religious beliefs. This is because they do not constitute an inviolable realm possessing absolute authority, as religious beliefs do, but can be revised or discarded at any time based on new experimental results and accumulated evidence. Furthermore, the criteria for judgment lie not in personal conviction but in rational reasoning and verifiable evidence. Therefore, the claim that mathematics became a branch of theology due to the Incompleteness Theorem is not valid; moreover, it can be argued that not only mathematics but science as well can be sufficiently independent of religious belief.
