Mathematical Foundations of Image Processing and Analysis, Volume 1

Elements of Mathematical Terminology

The purpose of this short, preliminary chapter is to introduce and define some of the key elements of general mathematical terminology which will be used in this book. Indeed, a clear and precise vocabulary is essential for a good understanding of concepts and ideas.

Axiom–Postulate–Conjecture–Paradox

An axiom is simply a premise or starting point from which other statements are logically derived. A postulate is a closely related notion, but slightly distinguished from the axiom, the latter will not seek to demonstrate. Euclidean Geometry is built from four axioms and a postulate (called the parallel postulate, or also Euclid’s fifth postulate that can be expressed as “At most one line can be drawn through any point not on a given line parallel to the given line in a plane”).

A theorem is a true statement that has been proven on the basis of previously established statements, and basically axioms/postulates.

A conjecture is an unproven assertion that appears to be true in the absence of a counter example, but which may appear doubtful for some people. In Number Theory, Fermat’s last theorem (also called Fermat’s conjecture) formulated in 1637 in a note at the margin of a copy of Diophantus’ Arithmetica, states that no three positive integers i, j, and k can satisfy the equation iⁿ + jⁿ = kⁿ for any integer value of n greater than two. Wiles’s proof was reported in 1995 [WIL 95].

A paradox is a statement that apparently contradicts itself and ...