What is a mathematical structure?

The notion of a mathematical structure was introduced at the beginning of XX century.

However, for a long time, mathematical objects were believed to be given always together with some structure, not necessarily unique, but at least natural (canonical). And only a practice, e.g., of functional analysis has led to conclusion that a canonical structure need not exist. For instance, there are different “natural” topologies of a set of rational numbers, different smooth structures of a 4-dimensional topological Euclidean space, different measures on a real line, and so on.

In mathematics, different types of structures are considered. These are an algebraic structure, a topological structure, cells whose notion generalizes the Boolean algebras and so on. In the first volume of their course, Bourbaki provide a description of a mathematical structure which enables them to define “espece de structure” and, thus, characterize and compare different structures. However, this is a structure of mathematical theories formulated in terms of logic. Therefore, one can suggest a wider definition of a structure which absorbs the Bourbaki one and the others, but can not characterize different types of structures.

This definition is based on the notion of a relation on a set and generalizes the definition of  a relational system in set theory.

Morphisms and functions are structures in this sense, and this fact provides a wide circle of applications of this notion of a structure to physics.