The book was an attempt to describe a set of axioms, inference rules and law of noncontradiction in symbolic logic from which all mathematical truths could in principle be proved. This ambitious project is of great importance in the history of mathematics and philosophy. The authors believed that such a project could be done. However, in 1931, Gödel's incompleteness theorem proved that "PM", and any other attempt, could never reach this goal. For any set of axioms and inference rules proposed, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.