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Rigour (BrE) or rigor (AmE) (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.
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. Consistency can be at odds here with a forgiving attitude, adaptability, and the need to take precedent with a pinch of salt. If a topic or case is dealt with in a rigorous way, it means that it is dealt with in a comprehensive, thorough and complete way, leaving no room for inconsistencies.
"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. It requires close attention to criteria for logical consistency, as well as to all relevant evidence and possible differences of interpretation.
In relation to intellectual honesty
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. 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.
Politics and the law
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).
In relation to mathematical proof
Mathematical rigour 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. During the 19th century, the term 'rigorous' began to be used to describe decreasing levels of abstraction when dealing with calculus which eventually became known as analysis. The works of Cauchy added rigour to the older works of Euler and Gauss. The works of Riemann added rigour to the works of Cauchy. The works of Weierstrass added rigour to the works of Riemann, eventually culminating in the arithmetization of analysis. Starting in the 1870s, the term gradually came to be associated with Cantorian set theory.
Mathematical rigour 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). 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 to a human mind may have fairly long formal derivations from the axioms. Under this argument, there is a trade-off between rigour and comprehension. Some argue that the use of formal languages to institute complete mathematical rigour might make theories which are commonly disputed or misinterpreted completely unambiguous by revealing flaws in reasoning.
In relation to physics
The role of mathematical rigour in relation to physics is twofold:
1. 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.
2. Second, there is the question regarding the role and status of mathematically rigorous results and relations[clarification needed]. This question is particularly vexing in relation to quantum field theory.
In relation to the classroom
Rigour in the classroom is a hotly debated topic amongst educators. Generally speaking, however, classroom rigour consists 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 developmental psychologist Jean Piaget, can build upon those skills with the help of a properly trained teacher.
Rigour in the classroom is commonly referred to as rigorous instruction. It is instruction that requires students to construct meaning for themselves, impose structure on information, integrate individual skills into processes, operate within but at the outer edge of their abilities, and apply what they learn in more than one context and to unpredictable situations 
- Bartlett, John, comp. Familiar Quotations, 10th ed, rev. and enl. by Nathan Haskell Dole. Boston: Little, Brown, 1919; Bartleby.com, 2000. http://www.bartleby.com/100/343.html. Retrieved Oct. 25, 2006.
- Wiener, N. (1985). Intellectual honesty and the contemporary scientist. In P. Masani (Ed.), Norbert Wiener: Collected works and commentary (pp. 725- 729).
- Hardware memory errors are caused by high-energy radiation from outer space, and can generally be expected to affect one bit of data per month, per gigabyte of DRAM..
- This refers to the 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner.
- Gelfert, Axel, 'Mathematical Rigor in Physics: Putting Exact Results in Their Place', Philosophy of Science, 72 (2005) 723-738.
- Forum: Academic Rigor, in: UNIversitas: The University of Northern Iowa Journal of Research, Scholarship, and Creative Activity 1.1 (Fall 2005).
- Jackson, R. (2011). How to Plan Rigorous Instruction. Alexandria, VA.: ASCD.