Rigour
Encyclopedia
Rigour or rigor 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.
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. 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 pedant
ry, 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.
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
.
, 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.
) 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.
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. 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 pedant
ry, 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.
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
.
, 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.
) 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.
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. 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 pedant
ry, 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.
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
.
, 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 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. Mathematical rigour in the modern sense originates no later than the generation of Bolzano
, Cauchy, and Abel
. Cauchy's starting point was the rejection of the principle of the generality of algebra
as practiced by Euler, Lagrange
, and other luminaries of earlier generations. Starting in the 1870s, the term gradually came to be associated with an implementation of the Weierstrass (ε, δ)-definition of limit and 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.
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
.
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. and works quoted therein.)
, according to developmental psychologist Jean Piaget
, can build upon those skills with the help of a properly trained teacher.
Religion
Religion is a collection of cultural systems, belief systems, and worldviews that establishes symbols that relate humanity to spirituality and, sometimes, to moral values. Many religions have narratives, symbols, traditions and sacred histories that are intended to give meaning to life or to...
, 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 standardDouble standard
A double standard is the unjust application of different sets of principles for similar situations. The concept implies that a single set of principles encompassing all situations is the desirable ideal. The term has been used in print since at least 1895...
be allowed: uniform principles should be applied. This is a test of consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
, 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
Precedent
In common law legal systems, a precedent or authority is a principle or rule established in a legal case that a court or other judicial body may apply when deciding subsequent cases with similar issues or facts...
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
Whist
Whist is a classic English trick-taking card game which was played widely in the 18th and 19th centuries. It derives from the 16th century game of Trump or Ruff, via Ruff and Honours...
. It implies that the demands of thinking accurately and to the point over a card game
Card game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games...
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 pedant
Pedant
A pedant is a person who is excessively concerned with formalism and precision, or who makes a show of his or her learning.-Etymology:The English language word "pedant" comes from the French pédant or its older mid-15th Century Italian source pedante, "teacher, schoolmaster"...
ry, which is intellectual rigour applied to no particular end, except perhaps self-importance.
Scholarship
Scholarly method
Scholarly method or scholarship is the body of principles and practices used by scholars to make their claims about the world as valid and trustworthy as possible, and to make them known to the scholarly public.-Methods:...
can be defined as intellectual rigour applied to the quality control
Quality control
Quality control, or QC for short, is a process by which entities review the quality of all factors involved in production. This approach places an emphasis on three aspects:...
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 validValidity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
evidence
Evidence
Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...
. 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
Debate
Debate or debating is a method of interactive and representational argument. Debate is a broader form of argument than logical argument, which only examines consistency from axiom, and factual argument, which only examines what is or isn't the case or rhetoric which is a technique of persuasion...
, if one wishes to say that an argument is flawed in its premises
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...
.
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 politicsPolitics
Politics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...
, for example. Arguing one way one day, and another later, can be defended by casuistry
Casuistry
In applied ethics, casuistry is case-based reasoning. Casuistry is used in juridical and ethical discussions of law and ethics, and often is a critique of principle- or rule-based reasoning...
, 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
In law, case law is the set of reported judicial decisions of selected appellate courts and other courts of first instance which make new interpretations of the law and, therefore, can be cited as precedents in a process known as stare decisis...
can therefore be at odds with a principled approach; and intellectual rigour can seem to be defeated. This defines a judge
Judge
A judge is a person who presides over court proceedings, either alone or as part of a panel of judges. The powers, functions, method of appointment, discipline, and training of judges vary widely across different jurisdictions. The judge is supposed to conduct the trial impartially and in an open...
'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
Rigour or rigor (see spelling differencesAmerican and British English spelling differences
One of the ways in which American English and British English differ is in spelling.-Historical origins:In the early 18th century, English spelling was not standardized. Differences became noticeable after the publishing of influential dictionaries...
) 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
Religion
Religion is a collection of cultural systems, belief systems, and worldviews that establishes symbols that relate humanity to spirituality and, sometimes, to moral values. Many religions have narratives, symbols, traditions and sacred histories that are intended to give meaning to life or to...
, 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 standardDouble standard
A double standard is the unjust application of different sets of principles for similar situations. The concept implies that a single set of principles encompassing all situations is the desirable ideal. The term has been used in print since at least 1895...
be allowed: uniform principles should be applied. This is a test of consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
, 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
Precedent
In common law legal systems, a precedent or authority is a principle or rule established in a legal case that a court or other judicial body may apply when deciding subsequent cases with similar issues or facts...
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
Whist
Whist is a classic English trick-taking card game which was played widely in the 18th and 19th centuries. It derives from the 16th century game of Trump or Ruff, via Ruff and Honours...
. It implies that the demands of thinking accurately and to the point over a card game
Card game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games...
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 pedant
Pedant
A pedant is a person who is excessively concerned with formalism and precision, or who makes a show of his or her learning.-Etymology:The English language word "pedant" comes from the French pédant or its older mid-15th Century Italian source pedante, "teacher, schoolmaster"...
ry, which is intellectual rigour applied to no particular end, except perhaps self-importance.
Scholarship
Scholarly method
Scholarly method or scholarship is the body of principles and practices used by scholars to make their claims about the world as valid and trustworthy as possible, and to make them known to the scholarly public.-Methods:...
can be defined as intellectual rigour applied to the quality control
Quality control
Quality control, or QC for short, is a process by which entities review the quality of all factors involved in production. This approach places an emphasis on three aspects:...
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 validValidity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
evidence
Evidence
Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...
. 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
Debate
Debate or debating is a method of interactive and representational argument. Debate is a broader form of argument than logical argument, which only examines consistency from axiom, and factual argument, which only examines what is or isn't the case or rhetoric which is a technique of persuasion...
, if one wishes to say that an argument is flawed in its premises
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...
.
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 politicsPolitics
Politics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...
, for example. Arguing one way one day, and another later, can be defended by casuistry
Casuistry
In applied ethics, casuistry is case-based reasoning. Casuistry is used in juridical and ethical discussions of law and ethics, and often is a critique of principle- or rule-based reasoning...
, 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
In law, case law is the set of reported judicial decisions of selected appellate courts and other courts of first instance which make new interpretations of the law and, therefore, can be cited as precedents in a process known as stare decisis...
can therefore be at odds with a principled approach; and intellectual rigour can seem to be defeated. This defines a judge
Judge
A judge is a person who presides over court proceedings, either alone or as part of a panel of judges. The powers, functions, method of appointment, discipline, and training of judges vary widely across different jurisdictions. The judge is supposed to conduct the trial impartially and in an open...
'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
Rigour or rigor (see spelling differencesAmerican and British English spelling differences
One of the ways in which American English and British English differ is in spelling.-Historical origins:In the early 18th century, English spelling was not standardized. Differences became noticeable after the publishing of influential dictionaries...
) 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
Religion
Religion is a collection of cultural systems, belief systems, and worldviews that establishes symbols that relate humanity to spirituality and, sometimes, to moral values. Many religions have narratives, symbols, traditions and sacred histories that are intended to give meaning to life or to...
, 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 standardDouble standard
A double standard is the unjust application of different sets of principles for similar situations. The concept implies that a single set of principles encompassing all situations is the desirable ideal. The term has been used in print since at least 1895...
be allowed: uniform principles should be applied. This is a test of consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
, 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
Precedent
In common law legal systems, a precedent or authority is a principle or rule established in a legal case that a court or other judicial body may apply when deciding subsequent cases with similar issues or facts...
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
Whist
Whist is a classic English trick-taking card game which was played widely in the 18th and 19th centuries. It derives from the 16th century game of Trump or Ruff, via Ruff and Honours...
. It implies that the demands of thinking accurately and to the point over a card game
Card game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games...
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 pedant
Pedant
A pedant is a person who is excessively concerned with formalism and precision, or who makes a show of his or her learning.-Etymology:The English language word "pedant" comes from the French pédant or its older mid-15th Century Italian source pedante, "teacher, schoolmaster"...
ry, which is intellectual rigour applied to no particular end, except perhaps self-importance.
Scholarship
Scholarly method
Scholarly method or scholarship is the body of principles and practices used by scholars to make their claims about the world as valid and trustworthy as possible, and to make them known to the scholarly public.-Methods:...
can be defined as intellectual rigour applied to the quality control
Quality control
Quality control, or QC for short, is a process by which entities review the quality of all factors involved in production. This approach places an emphasis on three aspects:...
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 validValidity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
evidence
Evidence
Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...
. 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
Debate
Debate or debating is a method of interactive and representational argument. Debate is a broader form of argument than logical argument, which only examines consistency from axiom, and factual argument, which only examines what is or isn't the case or rhetoric which is a technique of persuasion...
, if one wishes to say that an argument is flawed in its premises
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...
.
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 politicsPolitics
Politics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...
, for example. Arguing one way one day, and another later, can be defended by casuistry
Casuistry
In applied ethics, casuistry is case-based reasoning. Casuistry is used in juridical and ethical discussions of law and ethics, and often is a critique of principle- or rule-based reasoning...
, 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
In law, case law is the set of reported judicial decisions of selected appellate courts and other courts of first instance which make new interpretations of the law and, therefore, can be cited as precedents in a process known as stare decisis...
can therefore be at odds with a principled approach; and intellectual rigour can seem to be defeated. This defines a judge
Judge
A judge is a person who presides over court proceedings, either alone or as part of a panel of judges. The powers, functions, method of appointment, discipline, and training of judges vary widely across different jurisdictions. The judge is supposed to conduct the trial impartially and in an open...
'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
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
. It has a history traced back to Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
, in the work of Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
. This refers to the axiomatic method. Mathematical rigour in the modern sense originates no later than the generation of Bolzano
Bernard Bolzano
Bernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.-Family:Bolzano was the son of two pious Catholics...
, Cauchy, and Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
. Cauchy's starting point was the rejection of the principle of the generality of algebra
Generality of algebra
In the history of mathematics, the generality of algebra is phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph Lagrange...
as practiced by Euler, Lagrange
Lagrange
La Grange literally means the barn in French. Lagrange may refer to:- People :* Charles Varlet de La Grange , French actor* Georges Lagrange , translator to and writer in Esperanto...
, and other luminaries of earlier generations. Starting in the 1870s, the term gradually came to be associated with an implementation of the Weierstrass (ε, δ)-definition of limit and Cantorian
Cantor
Cantor is surname of:* Andrés Cantor , Spanish-language soccer announcer* Anthony Cantor , British diplomat* Arthur Cantor , American theatrical producer* Aviva Cantor , American journalist, lecturer and author...
set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
.
Mathematical rigour can be defined as amenability to algorithmic proof checking
Proof checking
Automated proof checking is the process of using software for checking proofs for correctness. It is one of the most developed fields in automated reasoning....
. 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
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
where such proofs can be codified using set theories such as ZFC (see automated theorem proving
Automated theorem proving
Automated theorem proving or automated deduction, currently the most well-developed subfield of automated reasoning , is the proving of mathematical theorems by a computer program.- Decidability of the problem :...
).
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.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
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
.
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
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
Both aspects of mathematical rigour in physics have attracted considerable attention in philosophy of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
. (See, for example, ref. and works quoted therein.)
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 developmentCognitive 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 brain development and cognitive psychology compared to an adult's point of...
, according to developmental psychologist Jean Piaget
Jean Piaget
Jean Piaget was a French-speaking Swiss developmental psychologist and philosopher known for his epistemological studies with children. His 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.
See also
- HonestyHonestyHonesty refers to a facet of moral character and denotes positive, virtuous attributes such as integrity, truthfulness, and straightforwardness along with the absence of lying, cheating, or theft....
- IntellectualIntellectualAn intellectual is a person who uses intelligence and critical or analytical reasoning in either a professional or a personal capacity.- Terminology and endeavours :"Intellectual" can denote four types of persons:...
- Intellectual dishonestyIntellectual dishonestyIntellectual dishonesty is dishonesty in performing intellectual activities like thought or communication. Examples are:* the advocacy of a position which the advocate knows or believes to be false or misleading...
- PedantPedantA pedant is a person who is excessively concerned with formalism and precision, or who makes a show of his or her learning.-Etymology:The English language word "pedant" comes from the French pédant or its older mid-15th Century Italian source pedante, "teacher, schoolmaster"...
- Scientific methodScientific methodScientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
- Self-deceptionSelf-deceptionSelf-deception is a process of denying or rationalizing away the relevance, significance, or importance of opposing evidence and logical argument...
- Sophistry
