Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse. These are separate from judicial and political applications with their suggestion of laws enforced to the letter, or political absolutism. A religion, too, may be worn lightly, or applied with rigour.
"The rigour of the game" is a quotation from Charles Lamb about whist. It implies that the demands of thinking accurately and to the point over a card game can serve also as entertainment or leisure. Intellectual rigour can therefore be sometimes seen as the exercise of a skill. It can also degenerate into pedantry, which is intellectual rigour applied to no particular end, except perhaps self-importance. Scholarship can be defined as intellectual rigour applied to the quality control of information, which implies an appropriate standard of accuracy, and scepticism applied to accepting anything on trust.
Mathematical rigour can be defined as amenability to algorithmic checking of correctness. Indeed, with the aid of computers, it is possible to check proofs mechanically by noting that possible flaws arise from either an incorrect proof or machine errors (which are extremely rare). Formal rigour is the introduction of high degrees of completeness by means of a formal language where such proofs can be codified using set theories such as ZFC (see automated theorem proving).
Most mathematical arguments are presented as prototypes of formally rigorous proofs. The reason often cited for this is that completely rigorous proofs, which tend to be longer and more unwieldy, may obscure what is being demonstrated. Steps which are obvious (as obvious as the axioms) to a human mind may have fairly long formal derivations from the axioms. Under this argument, there is a tradeoff between rigour and comprehension. Some argue that the utilisation of formal languages to institute complete mathematical rigour might make theories which are commonly disputed or misinterpreted, such as statistics, completely unambiguous.
The role of mathematical rigour in relation to physics is twofold.
First, there is the general question, sometimes called Wigner's Puzzle, "how it is that mathematics, quite generally, is applicable to nature?" However, scientists assume its successful application to nature justifies the study of mathematical physics.
Second, there is the question regarding the role and status of mathematically rigorous results and relations. This question is particularly vexing in relation to quantum field theory.
Both aspects of mathematical rigour in physics have attracted considerable attention in philosophy of science. (See, for example, ref. and works quoted therein.)
Rigor in the classroom is a hotly debated topic amongst educators. Generally speaking, however, classroom rigor is comprised of multi-faceted, challenging instruction and correct placement of the student. Students excelling in formal operational thought tend to excel in classes for gifted students. Students who have not reached that final stage of cognitive development, according to Piaget, can build upon those skills with the help of a properly trained teacher.