Editing Characterization (mathematics) (section)

{{Short description|Term in mathematics}}
In [[mathematics]], a '''characterization''' of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it.<ref name=":0">{{Cite web|url=http://mathworld.wolfram.com/Characterization.html|title=Characterization|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-21}}</ref> To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X'' have [[property (philosophy)|property]] ''P'', but that ''X'' is the ''only'' thing that has property ''P'' (i.e., ''P'' is a defining property of ''X''). Similarly, a set of properties ''P'' is said to characterize ''X'', when these properties distinguish ''X'' from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of ''X'' in terms of ''P'' include "''P'' is [[necessary and sufficient]] for ''X''", and "''X'' holds [[if and only if]] ''P''".

It is also common to find statements such as "Property ''Q'' characterizes ''Y'' [[up to]] [[isomorphism]]". The first type of statement says in different words that the [[extension (semantics)|extension]] of ''P'' is a [[singleton (mathematics)|singleton]] set, while the second says that the extension of ''Q'' is a single [[equivalence class]] (for isomorphism, in the given example &mdash; depending on how ''[[up to]]'' is being used, some other [[equivalence relation]] might be involved).

A reference on mathematical terminology notes that ''characteristic'' originates from the Greek term ''kharax'', "a pointed stake":<blockquote>From Greek ''kharax'' came ''kharakhter'', an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive nature. The Late Greek suffix ''-istikos'' converted the noun ''character'' into the adjective ''characteristic'', which, in addition to maintaining its adjectival meaning, later became a noun as well.<ref>Steven Schwartzmann (1994) ''The Words of Mathematics: An etymological dictionary of mathematical terms used in English'', page 43, [[The Mathematical Association of America]] {{ISBN|0-88385-511-9}}</ref></blockquote>Just as in chemistry, the [[characteristic property]] of a material will serve to identify a sample, or in the study of materials, structures and properties will determine [[characterization (materials science)|characterization]], in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in ''[[Mathematical Reviews]]'', as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review.<!-- Might consider a different reference, since the access to Mathematics Reviews requires active subscription . -->

In an arbitrary context of objects and features, characterizations have been expressed via the [[heterogeneous relation]] ''aRb'', meaning that object ''a'' has feature ''b''. For example, ''b'' may mean [[abstract and concrete|abstract or concrete]]. The objects can be considered the [[extension (semantics)|extension]]s of the world, while the features are expressions of the [[intension]]s. A continuing program of characterization of various objects leads to their [[categorization]].