29 April 21

Yesterday, 28 April, marked the birthday of mathematics-and-philosophy hero Kurt Gödel (1906-1978). He is most famous for developing the "incompleteness theorems," which say in simplest terms that our systems of mathematics and logic cannot prove themselves to be consistent, and so they are forever incomplete and "open-ended." I chose my tie based on M. C. Escher's artwork "Another World II" for today because its topsy-turvy picture of gravitation (in all directions) suggests how our perceptions are endlessly relative to our position in the universe. In the incompleteness theorems, Gödel was not telling us simply that numbers are endless, and (this is very important to understand) he was not saying that there are truths that literally cannot be proven. He means instead that the (excellent and trustworthy) rules we use for mathematics and logic are (simply) too limited to justify themselves (completely). From the viewpoint of philosophy, our rules for logic and math are constructed "by us" as instruments to understand the universe, and so it is not really surprising that they always provide "incomplete" interpretations of reality. Gödel's work shows that we should not despair about our limitations -- they are essential elements of anything worthy of being called a system.  Here comes the technical language: "Gödel’s two incompleteness theorems . . . concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent)."