This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia Mathematica.

During the Second World War, the Allies faced a question colloquially known as the “German Tank Problem”: how many tanks will the Axis ever produce? The answer resulted from an elegant probabilistic argument which was used by Allied mathematicians to make successful upper-bound estimates for the total Axis tank production. This paper shows that if two empirical postulates are true of the history of science, a parallel argument can be used to come up with lower-bound estimates for the number of (...) alternative scientific theories that remain undiscovered. The lower bound in question increases proportionally with the number of theories that have already been discovered. So, the problem of underconsideration is a serious problem and it will get worse, not better, as we discover new theories. (shrink)