Everything about Rigour totally explained
» For the medical term see rigor (medicine)
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.
Intellectual rigour
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 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.
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 isn't very rigorous, although very common in
politics, for example. Arguing one way one day, and another later, can be defended by
casuistry, for example 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
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, where it's said to have been invented. Complete rigour, it's often said, became available in
mathematics at the start of the
twentieth century. This of course refers to the
axiomatic method.
Mathematical rigour can be defined as amenability to algorithmic checking of correctness. Indeed, with the aid of computers, it's 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 rigourous proofs. The reason often cited for this is that completely rigourous 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's 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.
In relation to physics
The role of mathematical rigour in relation to physics is twofold.
First, there's the general question, sometimes called
Wigner's Puzzle, "how it's that mathematics, quite generally, is applicable to nature?" However, scientists assume its successful application to nature justifies the study of
mathematical physics.
Second, there's 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.)
In relation to the classroom
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 though 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.
Further Information
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