Abstraction by identification

A manner of forming general abstract concepts in which, when considering concrete initial objects, only those differences are considered that, for any reason, are relevant to the given situation, while other differences that are regarded as irrelevant are ignored. Initial objects differing from one another only in irrelevant aspects are considered as identical. Linguistically, abstraction by identification is manifested by the fact that two similar initial objects are identified with each other, and are spoken of as a single object, after having been assigned some suitable term. For instance, by identifying similar letters (words, alphabets) with each other, one arrives at the concept of an abstract letter, word or alphabet (see [1]); by identifying equivalent fundamental sequences of rational numbers one arrives at the concept of real numbers (cf. Real number); by identifying isomorphic groups with each other, one arrives at the concept of an abstract group, etc.

See also Abstraction, mathematical; Equivalence; Isomorphism.

#### References

[1] |  A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: Trudy Mat. Inst. Steklov. 42 (1954))

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[2] |  A.A. Markov, "On the logic of constructive mathematics" , Moscow (1972) (In Russian)

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From The Encyclopedia of Math: Abstraction by identification.

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