mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and related fields, an abstract structure is a way of describing a set of mathematical objects and the relationships between them, focusing on the essential rules and properties rather than any specific meaning or example.{{Cite book , last=Mac Lane , first=Saunders , title=Categories for the working mathematician , date=2010 , publisher=Springer , isbn=978-1-4419-3123-8 , edition=2nd. ed., Softcover version of original hardcover edition 1998 , series=Graduate texts in mathematics , location=New York, NY For example, in a game such as

chess Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...

, the rules of how the pieces move and interact define the structure of the game, regardless of whether the pieces are made of wood or plastic. Similarly, an abstract structure defines a framework of objects, operations, and relationships. These structures are studied in their own right, revealing fundamental mathematical principles. While a real-world object or computer program might ''represent, instantiate,'' or ''implement'' an abstract structure, the structure itself exists as an abstract concept, independent of any particular representation. This abstraction allows to see common patterns across seemingly different areas of mathematics and to apply the same reasoning and tools to analyze them. 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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

and

but in the fields that apply them, as

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

and

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

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

philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

(especially the philosophy of mathematics). An abstract structure has a richer structure than a

concept A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...

or an

idea In philosophy and in common usage, an idea (from the Greek word: ἰδέα (idea), meaning 'a form, or a pattern') is the results of thought. Also in philosophy, ideas can also be mental representational images of some object. Many philosophe ...

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

democracy Democracy (from , ''dēmos'' 'people' and ''kratos'' 'rule') is a form of government in which political power is vested in the people or the population of a state. Under a minimalist definition of democracy, rulers are elected through competitiv ...

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

melody A melody (), also tune, voice, or line, is a linear succession of musical tones that the listener perceives as a single entity. In its most literal sense, a melody is a combination of Pitch (music), pitch and rhythm, while more figurativel ...

is an abstract structure, but an orchestration 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, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

is an abstract structure, but the theory of continental drift 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 Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

. * A

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

is an abstract structure, but a

natural language A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...

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

Examples

Notes

See also