Mathematics

Characteristic features of modern mathematics includes:

1. Its subject matter consists not only of given quantitative relations and forms but of all possible (and, in general, variable) quantitative relations and interdependencies among magnitudes. In geometry, the focus is on both spatial relations and forms but also on all possible forms similar to spatial ones. In algebra, the focus is on various abstract systems of objects with all possible laws of operation on them. In analysis, not only magnitudes are considered as variables, but the very functions themselves. In a functional space, all the functions of a given type (all the possible interdependencies among the variables) are brought together.

2. The abstractions of mathematics are distinguished by three features: the concern with quantitative relations and spatial forms, abstracted from all other properties of objects; their occurrence in a sequence of increasing degrees of abstraction, going very much further in this direction than the abstractions of the other sciences; and their focus on abstract concepts and their interrelations.

3. The absence of privileged rank for any mathematical entities.

4. Isomorphism takes the place of identity within the mathematical process ensuring the non-ontological character of mathematics and hence its power, fidelity and polyvalence.

5. Its unity as a consequence of the existence of a common language and agreement on common elementary structures.