Rigour (American English: "rigor") 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.
Intellectual rigour is an important part, though not the whole, of intellectual honesty. For the latter, one should be questioning one's own assumptions, not merely applying them relentlessly if precisely. It is possible to doubt whether complete intellectual honesty exists — on the grounds that we, none of us, can entirely master our own presuppositions — without doubting that certain kinds of intellectual rigour are potentially available. The distinction certainly matters greatly in debate, if one wishes to say that an argument is flawed in its premises.
The setting for intellectual rigour does tend to assume a principled position from which to advance or argue. An opportunistic tendency to use any argument at hand is not very rigorous, if very common in politics, for example. Arguing one way one day, and another later, can be defended by casuistry, i.e. by saying the cases are different. In the legal context, for practical purposes, cases do always differ. Case law can therefore be at odds with a principled approach; and intellectual rigour can seem to be defeated. This defines a judge's problem with uncodified law. Codified law poses a different problem, of interpretation and adaptation of definite principles without losing the point; here applying the letter of the law, with all due rigour, may on occasion seem to undermine the principled approach.
An attempted short definition of intellectual rigour might be that no suspicion of double standard be allowed: uniform principles should be applied. This is a test of consistency, over cases, and to individuals or institutions (including the speaker, the speaker's country and so on). Consistency can be at odds here with a forgiving attitude, adaptability, and the need to take precedent with a pinch of salt.
Mathematical rigour is often cited as a kind of gold standard for mathematical proof. It has a history, being traced back to Greek mathematics, where it is said to have been invented. Complete rigour, it is often said, became available in mathematics at the start of the twentieth century. This relies on the axiomatic method, and the subsequent development of pure mathematics under the axiomatic umbrella. With the aid of computers, it is possible to check proofs mechanically; throwing the possible flaws back onto machine errors that are considered unlikely events. Indeed, mathematical rigour may be defined as amenability to algorithmic checking of correctness. Formal rigour is the introduction of high degrees of completeness by means of a formal language. Most mathematical arguments are presented as prototypes of formally rigorous proofs, on the grounds that too much formality may in fact obscure what is being said.