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Rigour (
British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English language in England, or, more broadl ...
) or rigor (
American English American English, sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. English is the most widely spoken language in the United States and in most circumstances ...
; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as
mathematical proofs A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
which must maintain
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
answers; or socially imposed, such as the process of defining
ethics Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concerns m ...
and
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
.


Etymology

"Rigour" comes to
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
through old French (13th c., Modern French '' rigueur'') meaning "stiffness", which itself is based on the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''rigorem'' (nominative ''rigor'') "numbness, stiffness, hardness, firmness; roughness, rudeness", from the
verb A verb () is a word (part of speech) that in syntax generally conveys an action (''bring'', ''read'', ''walk'', ''run'', ''learn''), an occurrence (''happen'', ''become''), or a state of being (''be'', ''exist'', ''stand''). In the usual descrip ...
''rigere'' "to be stiff". The
noun A noun () is a word that generally functions as the name of a specific object or set of objects, such as living creatures, places, actions, qualities, states of existence, or ideas.Example nouns for: * Living creatures (including people, alive, ...
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 te ...
, situations in which they are obligated to follow
church law Canon law (from grc, κανών, , a 'straight measuring rod, ruler') is a set of ordinances and regulations made by ecclesiastical authority (church leadership) for the government of a Christian organization or church and its members. It is th ...
exactly, and in which situations they can be more forgiving yet still considered moral. ''
Rigor mortis Rigor mortis ( Latin: ''rigor'' "stiffness", and ''mortis'' "of death"), or postmortem rigidity, is the third stage of death. It is one of the recognizable signs of death, characterized by stiffening of the limbs of the corpse caused by chemi ...
'' 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 philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (; Latin for " tdoes not follow") is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic syst ...
. 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 the subject as valid and trustworthy as possible, and to make them known to the scholarly public. It is the me ...
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 method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific me ...
, 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 may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet) A flyer (or flier) is a form of paper advertisement intended for wide distrib ...
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 ( peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. Peer revie ...
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 this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidenc ...
. 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 Debate is a process that involves formal discourse on a particular topic, often including a moderator and audience. In a debate, arguments are put forward for often opposing viewpoints. Debates have historically occurred in public meetings, a ...
, if one wishes to say that an argument is flawed in its
premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s.


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 (from , ) is the set of activities that are associated with making decisions in groups, or other forms of power relations among individuals, such as the distribution of resources or status. The branch of social science that studie ...
, for example. Arguing one way one day, and another later, can be defended by
casuistry In ethics, casuistry ( ) is a process of reasoning that seeks to resolve moral problems by extracting or extending theoretical rules from a particular case, and reapplying those rules to new instances. This method occurs in applied ethics and j ...
, 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 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 a le ...
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 presides over court proceedings, either alone or as a part of a panel of judges. A judge hears all the witnesses and any other evidence presented by the barristers or solicitors of the case, assesses the credibility ...
's problem with uncodified
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
. 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).


Proof

Mathematical rigour is often cited as a kind of gold standard for
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
. Its history traces back to
Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
, especially to
Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ge ...
's ''
Elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...
''. Until the 19th century, the treatise 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, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
(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 any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually cont ...
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 a ...
,
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 protra ...
,
Tarski's axioms Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (i.e., that part of Euclidean geometry that is formulab ...
). During the 19th century, the term "rigorous" began to be used to describe increasing levels of abstraction when dealing with
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
which eventually became known as
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
. The works of
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
added rigour to the older works of
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
and
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. The works of
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
added rigour to the works of Cauchy. The works of
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, 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 ...
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 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 mathematics, is mostly concer ...
. 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 consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...
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 ma ...
). 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 mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', 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 natural science that studies matter, its 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 which re ...
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 refers to the development of 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 developmen ...
. # 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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, 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 a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
(see, for example, ref. and ref. and the works quoted therein).


Education

Rigour in the classroom is a hotly debated topic amongst educators. 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.


See also

* Intellectual honesty *
Intellectual dishonesty Intellectual honesty is an applied method of problem solving, characterised by an unbiased, honest attitude, which can be demonstrated in a number of different ways: * One's personal beliefs or politics do not interfere with the pursuit of truth ...
*
Pedant A pedant is a person who is excessively concerned with formalism, accuracy and precision, or one who makes an ostentatious and arrogant show of learning. Etymology The English language word ''pedant'' comes from the French ''pédant'' (used ...
*
Scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific me ...
*
Self-deception Self-deception is a process of denying or rationalizing away the relevance, significance, or importance of opposing evidence and logical argument. Self-deception involves convincing oneself of a truth (or lack of truth) so that one does not revea ...
*
Sophistry A sophist ( el, σοφιστής, sophistes) was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught ...
* Cognitive rigor


References

{{Philosophical logic Philosophical logic *