Boarders on Nostr: Different types of usefulness of a field of mathematics: - One can use the field to ...
Different types of usefulness of a field of mathematics:
- One can use the field to construct an empirically adequate model in some scientific discipline (calculus in classical mechanics)
- One can reduce a class of problems from one field to another (commutative algebra in number theory, linear algebra in differential topology)
- The field is đđđđđđđĄđđŁđ to a whole new class of questions or mode of thinking (topos theory in Grothendieck's algebraic geometry, the formalization of real analysis in the 19th century)
- The field unifies previously disparate ideas allowing us to use insights from one domain in another (set theory/category theory)
- One can gain a brand new perspective more amenable with your mode of thinking or different from a previously dominant view (Galois theory to study polynomial equations, combinatorial species, the use of cohomology for matroids/graphs etc.)
- One can use the field to construct an empirically adequate model in some scientific discipline (calculus in classical mechanics)
- One can reduce a class of problems from one field to another (commutative algebra in number theory, linear algebra in differential topology)
- The field is đđđđđđđĄđđŁđ to a whole new class of questions or mode of thinking (topos theory in Grothendieck's algebraic geometry, the formalization of real analysis in the 19th century)
- The field unifies previously disparate ideas allowing us to use insights from one domain in another (set theory/category theory)
- One can gain a brand new perspective more amenable with your mode of thinking or different from a previously dominant view (Galois theory to study polynomial equations, combinatorial species, the use of cohomology for matroids/graphs etc.)