Rigour

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Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. [1] These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.

Contents

Etymology

"Rigour" comes to English through old French (13th c., Modern French 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". [2] 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, situations in which they are obligated to follow church law exactly, and in which situations they can be more forgiving yet still considered moral. [3] Rigor mortis 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. 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. [4]

Scholarly method 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, 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 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 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. [5] 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 premises.

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, 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.

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. Its history traces back to Greek mathematics, especially to Euclid's Elements . [6]

Until the 19th century, Euclid's Elements was seen as extremely rigorous and profound, but in the late 19th century, Hilbert (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). [7] 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 to address this gap in rigour found in the Elements (e.g., Hilbert's axioms, Birkhoff's axioms, Tarski's axioms).

During the 19th century, the term "rigorous" began to be used to describe increasing levels of abstraction when dealing with calculus which eventually became known as mathematical 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 modelled as amenability to algorithmic proof checking. Indeed, with the aid of computers, it is possible to check some proofs mechanically. [8] 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).

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 , 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 is twofold:

  1. First, there is the general question, sometimes called Wigner's Puzzle , [9] "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.
  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, 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 (see, for example, ref. [10] and ref. [11] 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.[ citation needed ] 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 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 [12]

See also

Related Research Articles

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

<span class="mw-page-title-main">Euclidean geometry</span> Mathematical model of the physical space

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

<span class="mw-page-title-main">Theorem</span> In mathematics, a statement that has been proved

In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique.

<span class="mw-page-title-main">Mathematical proof</span> Reasoning for mathematical statements

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 proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

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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 contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.

<span class="mw-page-title-main">Synthetic geometry</span> Geometry without using coordinates

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In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption.

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<span class="mw-page-title-main">Catholic probabilism</span>

In Catholic moral theology, probabilism provides a way of answering the question about what to do when one does not know what to do. Probabilism proposes that one can follow an authoritative opinion regarding whether an act may be performed morally, even though the opposite opinion is more probable. It was first formulated in 1577 by Bartholomew Medina, OP, who taught at Salamanca.

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

References

  1. ↑ "Definition of RIGOR". www.merriam-webster.com. Retrieved 2019-10-20.
  2. ↑ "Rigor – Etymology". etymonline.com. 2001–2014. Retrieved 2015-01-10.
  3. ↑ Amort, Eusebio. Theologia Moralis Inter Rigorem et Laxitatem Medi.
  4. ↑ "GA1: Intellectual Rigour - Southern Cross University". www.scu.edu.au. Retrieved 2019-10-20.
  5. ↑ Wiener, N. (1985). Intellectual honesty and the contemporary scientist. In P. Masani (Ed.), Norbert Wiener: Collected works and commentary (pp. 725–729).
  6. ↑ Pierpont, James (January 1928). "Mathematical rigor, past and present". Bulletin of the American Mathematical Society. 34 (1): 23–53. doi: 10.1090/S0002-9904-1928-04507-X . ISSN   0002-9904. S2CID   120096416.
  7. ↑ For more, see Euclidean geometry—19th century and non-Euclidean geometry.
  8. ↑ 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. .
  9. ↑ This refers to the 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner.
  10. ↑ Davey, Kevin, ”Is Mathematical Rigor Necessary in Physics?”, The British Journal for the Philosophy of Science, 54 (2003) 439–463.
  11. ↑ Gelfert, Axel, ”Mathematical Rigor in Physics: Putting Exact Results in Their Place”, Philosophy of Science, 72 (2005) 723–738.
  12. ↑ Jackson, R. (2011). How to Plan Rigorous Instruction. Alexandria, VA.: ASCD.
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