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. 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.
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?" Some scientists believe that its record of 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, 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. and ref. and the works quoted therein). == Education ==