Abstract structure
Encyclopedia
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.
is an abstract structure, but a recipe
is not, because it depends on the properties and quantities of its ingredients.
A simple melody
is an abstract structure, but an orchestration
is not, because it depends on the properties of particular instruments.
Euclidean geometry
is an abstract structure, but the theory of continental drift
is not, because it depends on the geology of the Earth
.
A formal language
is an abstract structure, but a natural language
is not, because its rules of grammar and syntax are open to debate and interpretation.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
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
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
but in the fields that apply them, as computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, and in the studies that reflect on them, as philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
and especially the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
. Indeed, modern mathematics has been defined in a very general sense as the study of abstract structures (by the Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
group: see discussion there, at algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
and also structure
Structure (category theory)
In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications...
).
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
Instantiation
Instantiation or instance*Philosophy:*A modern concept similar to participation in classical Platonism, see the Theory of Forms.*The principle of instantiation, the idea that in order for a property to exist, it must be had by some object or substance.*Universal instantiation and existential...
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
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
or an idea
Idea
In the most narrow sense, an idea is just whatever is before the mind when one thinks. Very often, ideas are construed as representational images; i.e. images of some object. In other contexts, ideas are taken to be concepts, although abstract concepts do not necessarily appear as images...
. 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
Democracy
Democracy is generally defined as a form of government in which all adult citizens have an equal say in the decisions that affect their lives. Ideally, this includes equal participation in the proposal, development and passage of legislation into law...
, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess.
Examples
A sorting algorithmSorting algorithm
In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order...
is an abstract structure, but a recipe
Recipe
A recipe is a set of instructions that describe how to prepare or make something, especially a culinary dish.-Components:Modern culinary recipes normally consist of several components*The name of the dish...
is not, because it depends on the properties and quantities of its ingredients.
A simple melody
Melody
A melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...
is an abstract structure, but an orchestration
Orchestration
Orchestration is the study or practice of writing music for an orchestra or of adapting for orchestra music composed for another medium...
is not, because it depends on the properties of particular instruments.
Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
is an abstract structure, but the theory of continental drift
Continental drift
Continental drift is the movement of the Earth's continents relative to each other. The hypothesis that continents 'drift' was first put forward by Abraham Ortelius in 1596 and was fully developed by Alfred Wegener in 1912...
is not, because it depends on the geology of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
.
A formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
is an abstract structure, but a natural language
Natural language
In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...
is not, because its rules of grammar and syntax are open to debate and interpretation.
See also
- Abstraction in computer scienceAbstraction (computer science)In computer science, abstraction is the process by which data and programs are defined with a representation similar to its pictorial meaning as rooted in the more complex realm of human life and language with their higher need of summarization and categorization , while hiding away the...
- Abstraction in generalAbstractionAbstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
- Abstraction in mathematicsAbstraction (mathematics)Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...
- Abstract objectAbstract objectAn abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta An abstract object is an...
- Deductive apparatus
- Formal scienceFormal scienceThe formal sciences are the branches of knowledge that are concerned with formal systems, such as logic, mathematics, theoretical computer science, information theory, systems theory, decision theory, statistics, and some aspects of linguistics....
s - Mathematical structureMathematical structureIn mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....