Where does category collide?

The highest and most general classes under which anything that exists can be placed, and the basic ways a thing can be spoken of: what it is, how much, what sort, where, when, and so on. Aristotle's Categories, in Ackrill's 1963 translation and commentary, sets out ten such headings, with substance as the primary one and the rest describing quantity, quality, relation, place, and more. Later Kant reused the word for the fixed forms the mind imposes on experience, but in both uses a category is a top-level division of reality or of thought. The referent is a supreme class of being or of predication reached by argument, not a rule of grammar, a structure of objects and arrows, or a grouping learned by a mind.[1]

Unit

One of the highest classes under which anything can be said to be

Recovery means

Not applicable

Measured by

Identified by philosophical analysis of the different ways a thing can be said to be, not by any instrument

A precise mathematical structure made of a collection of objects and, for each pair of objects, a collection of arrows between them that can be composed whenever their ends meet, with the composition being associative and each object carrying an identity arrow. Eilenberg and Mac Lane introduced the idea in their 1945 paper General theory of natural equivalences, so that whole mathematical settings could be compared by the arrows that map one into another rather than by their internal contents. The referent is a formal system of objects and morphisms defined by axioms, and it has nothing to do with a highest kind of being, a mental grouping, or a class of word forms; the objects here need not be objects in any everyday meaning.[2]

Unit

A collection of objects with composable, associative arrows and identities

Recovery means

Not applicable

Measured by

Specified abstractly by listing its objects, its morphisms, and the composition rules the axioms require

A grouping of objects, events, or ideas that a mind treats as the same kind, letting a person respond to a new instance based on what it shares with familiar ones. Rosch's 1975 study Cognitive representations of semantic categories showed that everyday categories are not neat boxes with sharp edges but are organized around a best or most typical example, so a robin counts as a more central bird than a penguin does. The referent is a psychological grouping held in a mind and measured by how people sort and judge instances, unlike a supreme class of being, a formal structure of arrows, or a dimension of grammar; here a category can have fuzzy borders and a graded centre.[3]

Unit

A grouping of instances a mind treats as one kind, often built around a typical example

Recovery means

Not applicable

Measured by

Studied through typicality ratings, the speed and errors of sorting instances, and which examples come to mind first

A dimension of grammatical contrast that a language forces its word forms to mark, such as number, tense, gender, or part of speech, with a fixed set of competing values like singular against plural. Corbett's 2000 book Number treats one such category in depth, showing how languages carve up the marking of one against more than one in strikingly different ways. The referent is a system of contrasting grammatical values built into a language and found by comparing word forms, not a highest kind of being, a structure of objects and arrows, or a grouping a mind learns from experience; the class here lives in the grammar, not in the world or the head.[4]

Unit

A dimension of grammatical contrast with a fixed set of values a word form must select

Recovery means

Not applicable

Measured by

Established by comparing word forms and how they pattern across a language's grammar