rigour

Rigour in mathematics refers to the quality of being extremely accurate and precise. This pursuit of accuracy has been a cornerstone of mathematical development, particularly since the Greeks, who transformed it from a practical discipline into a theoretical one based on proof.123

The quest for absolute certainty and the elimination of contradictions led to formalist programs, like Hilbert's, which aimed to establish the consistency of mathematics through its formal language. However, Gödel's incompleteness theorems demonstrated that proving the consistency of mathematics requires a system stronger than mathematics itself. Despite this, most pure mathematicians operate under a tacit formalist or Platonist view, believing in the existence of mathematical entities. The foundations of mathematics continue to evolve, with ongoing debates among mathematicians, logicians, and philosophers.45