“The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions.” DoneLanguageElementsStandardsUniversalFunctionMathematicsFoundationCategoriesConstruction Book:Mathematics, form and function Source: Mathematics, form and function
“. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.” ShouldWellsKindPhilosophyFormUnderstandingGoalDealsObjectsFitApproachFunctionRelationMathematicsFoundationObjectivesAlternativesOperationsCategoriesCompositionReplacedOften IsMembership Author:Saunders Mac Lane
