Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse. These are separate from public 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.
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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.
"The rigour of the game" is a quotation from Charles Lamb^{[1]} 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 selfimportance. 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 — which means keeping one's convictions in proportion to one's valid evidence.^{[2]} 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 no one can entirely master his or her 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, although 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, the facts of 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.
Mathematical rigour can refer both to rigorous methods of mathematical proof and to rigorous methods of mathematical practice (thus relating to other interpretations of rigour).
Mathematical rigor is often cited as a kind of gold standard for mathematical proof. It has a history traced back to Greek mathematics, in the work of Euclid. This refers to the axiomatic method. Complete rigor, it is often said, became available in mathematics at the start of the twentieth century.
Mathematical rigor can be defined as amenability to algorithmic proof checking. 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).^{[3]} Formal rigor 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 to a human mind may have fairly long formal derivations from the axioms. Under this argument, there is a tradeoff between rigor and comprehension. Some argue that the use of formal languages to institute complete mathematical rigor might make theories which are commonly disputed or misinterpreted completely unambiguous by revealing flaws in reasoning, such as a misuse of statistics.
The role of mathematical rigour in relation to physics is twofold.
First, there is the general question, sometimes called Wigner's Puzzle,^{[4]} "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.^{[5]} and works quoted therein.)
Rigour in the classroom is a hotly debated topic amongst educators. Generally speaking, however, classroom rigour consists of multifaceted, challenging instruction and correct placement of the student. Students excelling in formal operational thought tend to excel in classes for gifted students^{[citation needed]}. 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.
