Goedel's Incompleteness Theorems

Goedel's Incompleteness Theorems and Swinburne's Cumulative Case for Theism

James Baird

Kurt Goedel's demonstration of the incompleteness of formal mathematics has implications which have not been fully appreciated. I argue that, given the widely accepted understanding of scientific explanation as explanation of phenomena in terms of general laws and specific boundary conditions, Goedel's theorems show that, given certain highly plausible assumptions, humans can never discover a complete scientific explanation of the phenomena of human mathematical ability, and consequently of human psychology in general.

In two recent books, Roger Penrose has developed an intricate argument that 1) Goedel's theorems do indeed seem to have this implication for science as currently understood, and that therefore 2) the current understanding of science will have to be modified to include real processes which are essentially non-lawlike. I accept 1) but argue against 2) on several grounds, including those used by Daniel Dennett in Darwin's Dangerous Idea. I conclude that the gap in scientific explanation is real.

This gap is just the sort that has been exploited by Richard Swinburne in building his cumulative inductive case for theism using the probability calculus. I defend Swinburne against two or three criticisms and conclude that his methodology is sound. I then deploy his form of argument to show that the particular kind of psychology which humans possess is highly unlikely to exist with no explanation. Having already shown that this kind of psychology almost certainly has no scientific explanation, we are left with a good probability that it has a personal explanation and a reasonably high probability that it has an explanation in terms of the personal God of theism. Goedel's theorems therefore raise the overall probability of a Swinburne style case for theism.