Mathematical Incompleteness and Divine Ineffability Br. Matthew T. Warnez BH (bio) 1. Introduction1 Perhaps ever since the revelation of the Divine Name on Sinai, humans have acknowledged the power of self-referential definitions. The ancient Hebrews so esteemed God’s self-description—“I am who I am” (Ex 3:14)—that they eschewed its vocalization. Without subjecting the name to a philosophical analysis, they intuited that it captured something of divine ineffability.2 The type of self-reference of interest here is not produced by merely any use of the first-person pronoun. Rather, we have in mind self-referential statements which not only assign properties, but which prescribe a definition. Of course, self-referential definitions are no less powerful than they are hazardous. When used carelessly they run the risk of paradox, and Scripture itself knows of this paradoxical dimension, especially in the form of the so-called Epimenides paradox. Paul’s letter to Titus carries a notable example: “One of [the Cretans] said, ‘Cretans are always liars’” (Ti 1:12). St. Jerome also discovered the Epimenides paradox in Psalm 116:11, where he explained the problem as follows: [End Page 49] “I said in my alarm, Every man is a liar!” Is David telling the truth or is he lying? If it is true that every man is a liar, and David’s statement, “Every man is a liar” is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement: “Every man is a liar,” consequently is not true. Whatever way you turn the proposition, the conclusion is a contradiction.3 Semantic conundrums like these, including the even more vexing liar sentence (e.g., “this statement is false”), were a subject of periodic interest during the Middle Ages. In the nineteenth century, however, the advent of “formal logic” proved to be a tremendous aid to systematizing the study of logical paradoxes. Formal systems allowed acts of human reasoning to be expressed symbolically—removing human error from deductive procedures—and as such were able to pinpoint sources of self-contradiction. This symbolic approach became so effective that, by the early twentieth century, it was hoped that formal logic would ultimately provide an unassailable foundation for mathematics itself. At first, this “formalist program” set off with cheerful optimism under the inspiration of mathematician David Hilbert; it seemed to many that a watertight formalization of mathematics was within reach. In 1931, however, Hilbert’s program was taken by surprise. An Austrian logician, Kurt Gödel, discovered that many useful formal systems of logic were capable of reproducing, within the formal systems themselves, a statement analogous to the liar sentence. This statement—called the Gödel sentence of a theory—is a sentence that asserts its own unprovability, similar to the way the liar sentence asserts its own falsity. But while the liar sentence generates a paradox, the Gödel sentence leads to two theorems—Gödel’s incompleteness theorems—which assert that formal deduction processes have striking limitations.4 Many outlandish claims have been made on behalf of these strictly mathematical theorems, ranging from proofs of the relativity of truth to proofs of objective reality.5 Indeed, among mathematical results, [End Page 50] the incompleteness theorems are uniquely susceptible to philosophical misappropriation, and great caution is warranted whenever they are displaced from symbolic logic.6 Nevertheless, the incompleteness theorems offer an epistemological perspective which, if applied judiciously, is itself valuable, regardless of its formal relation to symbolic logic. Therefore, proceeding cautiously, I will undertake to raise potential connections between the incompleteness theorems and aspects of the Christian worldview. After introducing the mathematical content of the incompleteness theorems, I will discuss several areas where the theorems seem to intersect with nature, man, and God. Finally, I will assess the legitimacy, in general, of theological interpretations of the incompleteness theorems—including the tantalizing correspondence between mathematical incompleteness and divine ineffability. I conclude, however, that the most valuable import of the incompleteness theorems seems to be found within their proper context, where the theorems epitomize the limits of human reason. 2. Mathematical Incompleteness Rather than presenting a detailed exposition of the incompleteness theorems here...
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Jun 19, 2024
David Worsley 
Open Access
