The Outer Limits: In Search of the “Unknowable” in Science

I. Logical Barriers in Science To anyone infected with the idea that the human mind is unlimited in its capacity to answer questions about natural and human affairs, a tour of twentieth-century science must be quite a depressing experience. Many of the deepest and most well-chronicled results of science in this century have been statements about what “cannot” be done and what “cannot” be known. Probably the most famous limitative result of this kind is Gödel’s Incompleteness Theorem, which tells us that no system of deductive inference is capable of answering all questions about numbers that can be stated using the language of the system. In short, every sufficiently powerful, consistent logical system is incomplete. A few years later, Alan Turing proved an equivalent assertion about computer programs, which states that there is no systematic way of testing a program and its data to say whether or not the program will ever halt when processing that data. More recently, Gregory Chaitin has looked at Gödel’s notion of provability from an information-theoretic perspective, finding explicit examples of simple arithmetic propositions whose truth or falsity will never be known by following the deductive rules of any system of logical inference. Essentially, what Chaitin’s results show is that such mathematical questions are simply too complex for us.