Abstraction, mathematical

Abstraction in mathematics, or mental abstraction, is a significant component of the mental activity aimed at the formulation of basic mathematical concepts. The most typical abstractions in mathematics are "pure" abstractions, idealizations and their various multi-layered superpositions (see [5]).

The mental act of "pure abstractionpure" abstraction consists of the fact that in a certain situation being considered, attention is fixed only on certain (essential) features of the objects being considered and on the interrelationships between them, with the exclusion of other properties and relationships which are considered as irrelevant. The result of such an act of abstraction, firmed in an appropriate language, begins to play the part of a general concept. A typical example of mathematical abstraction of this kind is abstraction by identification.

The mental act of idealization means that, in a certain situation being considered, one's imagination generates a certain concept which becomes the object considered by one's consciousness. The properties imparted to the concept by one's imagination are not only those actually possessed by the initial objects as a result of the act of "pure" abstraction, but also other, imagined properties, which reflect the original properties of the initial objects in a modified manner, or even properties altogether absent in reality. One of the most traditional mathematical idealizations is the abstraction of actual infinity, which leads to the idea of an actual infinity. This abstraction is the base of the set-theoretic development of mathematics. Another traditional idealization — the abstraction of potential realizability — leads to the idea of a potential infinity. This abstraction, in conjunction with the refusal to use the abstraction of actual infinity, forms the base of constructive foundations of mathematics.

The nature of any mathematical theory is largely determined by the nature of the mathematical abstraction on which the formulation of the fundamental concepts of this theory is based. The analysis of such abstractions is one of the principal tasks of the foundations of mathematics. A careful consideration of the problems related to this circle of questions resulted in the recognition of the fundamental importance of the following factors: 1) the evaluation of abstract objects resulting from the superposition of far-reaching idealizations requires the development of special means for their understanding; such a development is a difficult task, which forms the subject of a special discipline: semantics; and 2) the logical apparatus which may be applied to any given mathematical theory essentially depends on the nature of the fundamental concepts of this theory, and hence on the nature of the mathematical abstractions accepted in the formulation of these concepts. See Intuitionism; Constructive mathematics.

Major contributions to the analysis of abstractions used in mathematics were made by L.E.J. Brouwer [1], H. Weyl [2], D. Hilbert [3], A.A. Markov [4], and others.

References

[1]	L.E.J. Brouwer, "De onbetrouwbaarheid der logische principes" Tijdschrift voor Wijsbegeerte , 2 (1908) pp. 152–158
[2]	H. Weyl, "Das Kontinuum" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[3]	D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) pp. Appendix VIII
[4]	A.A. Markov, "On the logic of constructive mathematics" , Moscow (1972) (In Russian)
[5]	N.A. Shanin, "Constructive real numbers and constructive function spaces" Trudy Mat. Inst. Steklov. , 67 (1962) pp. 15–294 (In Russian)