An abstract structure in mathematics is a formal object that is defined by a set of laws, properties, and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects. Abstract structures are studied not only in logic and mathematics but in the fields that apply them, as computer science, and in the studies that reflect on them, as philosophy and especially the philosophy of mathematics. Indeed, modern mathematics has been defined in a very general sense as the study of abstract structures (by the Bourbaki group: see discussion there, at algebraic structure and also structure).
An abstract structure may be represented (perhaps with some degree of approximation) by one or more physical objects — this is called an implementation or instantiation of the abstract structure. But the abstract structure itself is defined in a way that is not dependent on the properties of any particular implementation.
An abstract structure has a richer structure than a concept or an idea. An abstract structure must include precise rules of behaviour which can be used to determine whether a candidate implementation actually matches the abstract structure in question. Thus we may debate how well a particular government fits the concept of democracy, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess.
Read more about Abstract Structure: Examples
Famous quotes containing the words abstract and/or structure:
“The reader uses his eyes as well as or instead of his ears and is in every way encouraged to take a more abstract view of the language he sees. The written or printed sentence lends itself to structural analysis as the spoken does not because the reader’s eye can play back and forth over the words, giving him time to divide the sentence into visually appreciated parts and to reflect on the grammatical function.”
—J. David Bolter (b. 1951)
“There is no such thing as a language, not if a language is anything like what many philosophers and linguists have supposed. There is therefore no such thing to be learned, mastered, or born with. We must give up the idea of a clearly defined shared structure which language-users acquire and then apply to cases.”
—Donald Davidson (b. 1917)