An abstract structure is an abstraction that might be of the

geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

spaces or a set structure, or a

hypostatic abstraction Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal operation that transforms a predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (phil ...

that is defined by a set of mathematical theorems and laws, properties and relationships in a way that is logically if not always historically independent However historical dependencies are partially considered in event theory as part of the

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

theory in

Kolmogorov complexity In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produ ...

and Kolmogorov-Khinchin equations of the structure of contingent experiences, for example, those involving physical objects. Abstract structures are studied not only in

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

and

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

but in the fields that apply them, as

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

and

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...

, and in the studies that reflect on them, such as

philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

(especially the

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...

). 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 In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...

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 Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by s ...

or an

idea In common usage and in philosophy, ideas are the results of thought. Also in philosophy, ideas can also be mental representational images of some object. Many philosophers have considered ideas to be a fundamental ontological category of being ...

. 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, and it must be free from

contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...

s. Thus we may debate how well a particular government fits the concept of

democracy Democracy (From grc, δημοκρατία, dēmokratía, ''dēmos'' 'people' and ''kratos'' 'rule') is a form of government in which the people have the authority to deliberate and decide legislation (" direct democracy"), or to choose gov ...

, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess (for example

Kasparov Garry Kimovich Kasparov (born 13 April 1963) is a Russian chess grandmaster, former World Chess Champion, writer, political activist and commentator. His peak Elo rating system, rating of 2851, achieved in 1999, was the highest recorded until ...

ian approaches).

Examples

* A

sorting algorithm In computer science, a sorting algorithm is an algorithm that puts elements of a List (computing), list into an Total order, order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. ...

is an abstract structure, but a

recipe A recipe is a set of instructions that describes how to prepare or make something, especially a dish of prepared food. A sub-recipe or subrecipe is a recipe for an ingredient that will be called for in the instructions for the main recipe. His ...

is not, because it depends on the properties and quantities of its ingredients. * A simple

melody A melody (from Greek language, Greek μελῳδία, ''melōidía'', "singing, chanting"), also tune, voice or line, is a Linearity#Music, linear succession of musical tones that the listener perceives as a single entity. In its most liter ...

is an abstract structure, but an

orchestration Orchestration is the study or practice of writing music for an orchestra (or, more loosely, for any musical ensemble, such as a concert band) or of adapting music composed for another medium for an orchestra. Also called "instrumentation", orc ...

is not, because it depends on the properties of particular instruments. *

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

is an abstract structure, but the theory of

continental drift Continental drift is the hypothesis that the Earth's continents have moved over geologic time relative to each other, thus appearing to have "drifted" across the ocean bed. The idea of continental drift has been subsumed into the science of pla ...

is not, because it depends on the geology of the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

. * A

formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symb ...

is an abstract structure, but a

natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

is not, because its rules of grammar and syntax are open to debate and interpretation.

Examples

Notes

See also