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.
