Argument
Isomorphism
Overview
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as
A
≅
B
{\displaystyle A\cong B}
. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.
The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified.
An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a universal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every prime number p, all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.
The term isomorphism is mainly used for algebraic structures and categories. In the case of algebraic structures, mappings are called homomorphisms, and a homomorphism is an isomorphism if and only if it is bijective.
In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:
An isometry is an isomorphism of metric spaces.
A homeomorphism is an isomorphism of topological spaces.
A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.
A symplectomorphism is an isomorphism of symplectic manifolds.
A permutation is an automorphism of a set.
In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.
Category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Let us take a series of elements at successive distances _x_ such that the distance between any two is the sum of the distances between intermediate elements. Such a series corresponds to a straight line, which is taken as the _x_-axis. Then a series _OY_ is called perpendicular to the _x_-axis _OX_, when the distances of any element _y_, on _OY_, from +_mx_ and -_mx_ are equal. By our hypothesis, these distances…
Bertrand Russell, An essay on the foundations of geometry
States what Isomorphism is; the author is tied to Isomorphism in the knowledge graph.
The logical basis of the theory alone belongs to philosophy, and alone will occupy us to-night. The way the problem of continuity enters into philosophy is, broadly speaking, the following: Space and time are treated by mathematicians as consisting of points and instants, but they also have a property, easier to feel than to define, which is called continuity, and is thought by many philosophers to be destroyed when…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Isomorphism is; the author is tied to Isomorphism in the knowledge graph.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Isomorphism is.
Related ideas
- Parity (mathematics)
- Integer
- If and only if
- Ludwig Wittgenstein
- Algebra
- Mathematics
- Geometry
- Group (mathematics)
- Mathematical object
- Rational number
- Reflexive relation
- Real number
- Cybernetics
- Partially ordered set
- Set (mathematics)
- Morphism
- Total order
- Topological space
- Bijection
- Type theory
- Bertrand Russell
- Mathematical analysis
- Map (mathematics)
- Category theory
Place in the tradition
Part of Functionalism (philosophy of mind).