Rigour

Rigour (British English) or rigor (American English; 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 which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.

Rigour (British English) or rigor (American English; 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 which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law. == Mathematics == Mathematical rigour can apply to methods of mathematical proof and to methods of mathematical practice (thus relating to other interpretations of rigour). 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. 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). 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. 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.

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