Rigour (

British English British English is the set of Variety (linguistics), varieties of the English language native to the United Kingdom, especially Great Britain. More narrowly, it can refer specifically to the English language in England, or, more broadly, to ...

) or rigor (

American English American English, sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States. English is the Languages of the United States, most widely spoken lang ...

; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of

famine A famine is a widespread scarcity of food caused by several possible factors, including, but not limited to war, natural disasters, crop failure, widespread poverty, an Financial crisis, economic catastrophe or government policies. This phenom ...

"; logically imposed, such as mathematical proofs which must maintain

consistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...

answers; or socially imposed, such as the process of defining

ethics Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...

and

law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a science and as the ar ...

Etymology

"Rigour" comes to English through

old French Old French (, , ; ) was the language spoken in most of the northern half of France approximately between the late 8th French '' rigueur'') meaning "stiffness", which itself is based on the Latin">Wiktionary:fr:rigueur">rigueur'') meaning "stiffness", which itself is based on the Latin ''rigorem'' (nominative ''rigor'') "numbness, stiffness, hardness, firmness; roughness, rudeness", from the verb ''rigere'' "to be stiff". The noun was frequently used to describe a condition of strictness or stiffness, which arises from a situation or constraint either chosen or experienced passively. For example, the title of the book ''Theologia Moralis Inter Rigorem et Laxitatem Medi'' roughly translates as "mediating theological morality between rigour and laxness". The book details, for the

clergy Clergy are formal leaders within established religions. Their roles and functions vary in different religious traditions, but usually involve presiding over specific rituals and teaching their religion's doctrines and practices. Some of the ter ...

, situations in which they are obligated to follow church law exactly, and in which situations they can be more forgiving yet still considered moral. ''

Rigor mortis Rigor mortis (), or postmortem rigidity, is the fourth stage of death. It is one of the recognizable signs of death, characterized by stiffening of the limbs of the corpse caused by chemical changes in the muscles postmortem (mainly calcium ...

'' translates directly as the stiffness (''rigor'') of death (''mortis''), again describing a condition which arises from a certain constraint (death).

Intellectualism

Intellectual rigour is a process of thought which is consistent, does not contain self-contradiction, and takes into account the entire scope of available knowledge on the topic. It actively avoids

logical fallacy In logic and philosophy, a formal fallacy is a pattern of reasoning rendered invalid by a flaw in its logical structure. Propositional logic, for example, is concerned with the meanings of sentences and the relationships between them. It focuses ...

. Furthermore, it requires a sceptical assessment of the available knowledge. If a topic or case is dealt with in a rigorous way, it typically means that it is dealt with in a comprehensive, thorough and complete way, leaving no room for inconsistencies.

Scholarly method The scholarly method or scholarship is the body of principles and practices used by scholars and academics to make their claims about their subjects of expertise as valid and trustworthy as possible, and to make them known to the scholarly pub ...

describes the different approaches or methods which may be taken to apply intellectual rigour on an institutional level to ensure the quality of information published. An example of intellectual rigour assisted by a methodical approach is the

scientific method The scientific method is an Empirical evidence, empirical method for acquiring knowledge that has been referred to while doing science since at least the 17th century. Historically, it was developed through the centuries from the ancient and ...

, in which a person will produce a hypothesis based on what they believe to be true, then construct experiments in order to prove that hypothesis wrong. This method, when followed correctly, helps to prevent against

circular reasoning Circular reasoning (, "circle in proving"; also known as circular logic) is a fallacy, logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect ...

and other fallacies which frequently plague conclusions within academia. Other disciplines, such as philosophy and mathematics, employ their own structures to ensure intellectual rigour. Each method requires close attention to criteria for logical consistency, as well as to all relevant evidence and possible differences of interpretation. At an institutional level,

peer review Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (:wiktionary:peer#Etymology 2, peers). It functions as a form of self-regulation by qualified members of a profession within the ...

is used to validate intellectual rigour.

Honesty

Intellectual rigour is a subset of intellectual honesty—a practice of thought in which ones convictions are kept in proportion to valid

evidence Evidence for a proposition is what supports the proposition. It is usually understood as an indication that the proposition is truth, true. The exact definition and role of evidence vary across different fields. In epistemology, evidence is what J ...

. Intellectual honesty is an unbiased approach to the acquisition, analysis, and transmission of ideas. A person is being intellectually honest when he or she, knowing the truth, states that truth, regardless of outside social/environmental pressures. 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

premise A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion. An argument is meaningf ...

Politics and 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 Politics () is the set of activities that are associated with decision-making, making decisions in social group, groups, or other forms of power (social and political), power relations among individuals, such as the distribution of Social sta ...

, for example. Arguing one way one day, and another later, can be defended by

casuistry Casuistry ( ) is a process of reasoning that seeks to resolve moral problems by extracting or extending abstract rules from a particular case, and reapplying those rules to new instances. This method occurs in applied ethics and jurisprudence. ...

, i.e. by saying the cases are different. In the legal context, for practical purposes, the facts of cases do always differ.

Case law Case law, also used interchangeably with common law, is a law that is based on precedents, that is the judicial decisions from previous cases, rather than law based on constitutions, statutes, or regulations. Case law uses the detailed facts of ...

can therefore be at odds with a principled approach; and intellectual rigour can seem to be defeated. This defines a

judge A judge is a person who wiktionary:preside, presides over court proceedings, either alone or as a part of a judicial panel. In an adversarial system, the judge hears all the witnesses and any other Evidence (law), evidence presented by the barris ...

's problem with uncodified

. 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''.

Mathematics

Mathematical rigour can apply to methods of mathematical proof and to methods of mathematical practice (thus relating to other interpretations of rigour).

Mathematical proof

Mathematical rigour is often cited as a kind of gold standard for

mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...

. Its history traces back to

Greek mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities ...

, especially to

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

's '' Elements''. Until the 19th century, Euclid's ''Elements'' was seen as extremely rigorous and profound, but in the late 19th century,

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...

(among others) realized that the work left certain assumptions implicit—assumptions that could not be proved from Euclid's Axioms (e.g. two circles can intersect in a point, some point is within an angle, and figures can be superimposed on each other). This was contrary to the idea of rigorous proof where all assumptions need to be stated and nothing can be left implicit. New foundations were developed using the

axiomatic method In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establis ...

to address this gap in rigour found in the ''Elements'' (e.g.,

Hilbert's axioms Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book ''Grundlagen der Geometrie'' (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of Euclidean geometry. Other well-known modern ax ...

Birkhoff's axioms In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and prot ...

, Tarski's axioms). During the 19th century, the term "rigorous" began to be used to describe increasing levels of abstraction when dealing with

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

which eventually became known as

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

. The works of Cauchy added rigour to the older works of Euler and

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

. The works of Riemann added rigour to the works of Cauchy. The works of

Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...

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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

. Mathematical rigour can be modelled as amenability to algorithmic proof checking. Indeed, with the aid of computers, it is possible to check some proofs mechanically. Formal rigour is the introduction of high degrees of completeness by means of a

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

where such proofs can be codified using set theories such as ZFC (see

automated theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a majo ...

). Published mathematical arguments have to conform to a standard of rigour, but are written in a mixture of symbolic and natural language. In this sense, written mathematical discourse is a prototype of formal proof. Often, a written proof is accepted as rigorous although it might not be formalised as yet. The reason often cited by mathematicians for writing informally is that completely formal proofs tend to be longer and more unwieldy, thereby obscuring the line of argument. An argument that appears obvious to human intuition may in fact require fairly long formal derivations from the axioms. A particularly well-known example is how in ''

Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...

'', Whitehead and Russell have to expend a number of lines of rather opaque effort in order to establish that, indeed, it is sensical to say: "1+1=2". In short, comprehensibility is favoured over formality in written discourse. Still, advocates of automated theorem provers may argue that the formalisation of proof does improve the mathematical rigour by disclosing gaps or flaws in informal written discourse. When the correctness of a proof is disputed, formalisation is a way to settle such a dispute as it helps to reduce misinterpretations or ambiguity.

Physics

The role of mathematical rigour in relation to

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

is twofold: # First, there is the general question, sometimes called '' Wigner's Puzzle'', "how it is that mathematics, quite generally, is applicable to nature?" Some scientists believe that its record of successful application to nature justifies the study of

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

. # 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

, where computations often produce infinite values for which a variety of non-rigorous work-arounds have been devised. Both aspects of mathematical rigour in physics have attracted considerable attention in

philosophy of science Philosophy of science is the branch of philosophy concerned with the foundations, methods, and implications of science. Amongst its central questions are the difference between science and non-science, the reliability of scientific theories, ...

(see, for example, ref. and ref. and the works quoted therein).

Education

Rigour in the classroom is a hotly debated topic amongst educators. Even the semantic meaning of the word is contested. Generally speaking, 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 Cognitive development is a field of study in neuroscience and psychology focusing on a child's development in terms of information processing, conceptual resources, perceptual skill, language learning, and other aspects of the developed adult bra ...

, according to developmental psychologist

Jean Piaget Jean William Fritz Piaget (, ; ; 9 August 1896 – 16 September 1980) was a Swiss psychologist known for his work on child development. Piaget's theory of cognitive development and epistemological view are together called genetic epistemology. ...

, can build upon those skills with the help of a properly trained teacher. Rigour in the classroom is commonly called "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.Jackson, R. (2011). ''How to Plan Rigorous Instruction''. Alexandria, VA.: ASCD.

References

{{Philosophical logic Philosophical logic *