List of Mathematical Jargon - Proof Techniques

Proof Techniques

Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for filling in tedious details.

angle chasing

Used to describe a geometrical proof that involves finding relationships between the various angles in a diagram.

back-of-the-envelope calculation

An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case.

by inspection

A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction. If an expression can be evaluated by straightforward application of simple techniques and without recourse to extended calculation or general theory, then it can be evaluated by inspection. It is also applied to solving equations; for example to find roots of a quadratic equation by inspection is to 'notice' them, or mentally check them. 'By inspection' can play a kind of gestalt role: the answer or solution simply clicks into place.

clearly, can be easily shown

A term which shortcuts around calculation the mathematician perceives to be tedious or routine, accessible to any member of the audience with the necessary expertise in the field; Laplace used obvious (French: évident).

diagram chasing

Given a commutative diagram of objects and morphisms between them, if one wishes to prove some property of the morphisms (such as injectivity) which can be stated in terms of elements, then the proof can proceed by tracing the path of elements of various objects around the diagram as successive morphisms are applied to it. That is, one chases elements around the diagram, or does a diagram chase.

handwaving

A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary. It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.

in general

In a context not requiring rigor, this phrase often appears as a labor-saving device when the technical details of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simple enough case that the computations are reasonable, and then indicates that "in general" the proof is similar.

morally true

Used to indicate that the speaker believes a statement should be true, given their mathematical experience, even though a proof has not yet been put forward. As a variation, the statement may in fact be false, but instead provide a slogan for or illustration of a correct principle. Hasse's local-global principle is a particularly influential example of this.

trivial

Similar to clearly. A concept is trivial if it holds by definition, is immediately corollary to a known statement, or is a simple special case of a more general concept.