User:D.Lazard/Lead for Mathematical object

"Mathematical object", or simply "object" is a colloquial term used in mathematics for anything that has been (or could be) defined in mathematical terms, and whose properties may be deduced with mathematical proofs. Typically, a mathematical object can be assigned to a variable, can be quantified, and therefore can be involved in formulas. Common mathematical objects include numbers, sets, mathematical structures such as field and spaces, functions, expressions, geometric objects, and transformations. Mathematical objects can be very complex; for example, in mathematical logic, theorems, proofs, and even theories are considered as mathematical objects.

Being a colloquial term, there is no formal definition of the concept, and it may depend on the author and the context whether a mathematical entity is considered as a mathematical object. For example, arithmetic operations are not generally considered as mathematical objects when used for computing, but are when studied as bivariate functions or ternary relations.

The term "mathematical object" was introduced in the 20th century, with the generalization of the use of set theory and the axiomatic method, which led to assign to variables and to manipulate new "objects" such as infinite sets, algebraic structures and spaces of various nature.

The objects of a category are mathematical objects, but many mathematical objects, such as numbers, are not objects of any category.

Mathematical objects are weakly related with abstract objects of philosophy: mathematical objects are abstract objects if one accept mathematical Platonism, but the concept of abstract object is much wider than that of mathematical object.

Historical motivation

Before the 17th century, the mathematical objects that were considered were essentially numbers and geometric shapes. As, at that time, there was no relation between these two sorts of entities, there was no need of a common term for them.

After the introduction of calculus new sorts of mathematical objects were introduced, such as function (mathematics), matrices and vectors. Again, different these different sorts of entities were totally distincts