On Definitions VII

"No step should ever be taken, and no statement should ever be made, without giving a legitimate reason therefor. A 'legitimate' reason means a reference to a definition, to an assumption, or to a proposition previously proved; it may also include reference to a statement in the hypothesis, or to a line arbitrarily drawn at the outset of the demonstration." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"To define a term adequately means to describe it in such a way that no doubt or ambiguity arises as to its meaning when that term is used. A good definition should indicate to what larger group of objects the object defined belongs, as well as how that object differs from other objects of that group. The larger group is sometimes called the genus, and the smaller, special group is called the species. Geometric objects, i.e., figures, are for the most part easy to define with precision." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"The definition of a problem and the action taken to solve it largely depend on the view which the individuals or groups that discovered the problem have of the system to which it refers. A problem may thus find itself defined as a badly interpreted output, or as a faulty output of a faulty output device, or as a faulty output due to a malfunction in an otherwise faultless system, or as a correct but undesired output from a faultless and thus undesirable system. All definitions but the last suggest corrective action; only the last definition suggests change, and so presents an unsolvable problem to anyone opposed to change." (Herbert Brün, "Technology and the Composer", 1971)

"A definition in mathematics is an exercise in uncovering the essence of things, one reason that good definitions are so hard to pull off, since a definition brings the essence to light, and the light brings the definition to life." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Definitions pin things down, they limit the prospects for creativity and diversity. A definition, implicitly, attempts to reduce all possible variations of a concept to a single pithy phrase." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Whenever you create or define a mathematical object, it always carries with it the blueprint of its own construction - the defining features that make it what it is and not some other thing." (Paul Lockhart, "Measurement", 2012)