What became of antiquity?
3 great problems solved
parallel postulate independent
geometry, arithmetic, logic all intimately connected now,
at a high level
modern day analogs:
- geometry: narrowly: differential geometry; broadly: geometry
- arithmetic: narrowly: algebraic number theory; broadly, algebra and number theory
- logic: narrowly: logic; broadly: logic, set theory, computation
exhaustion -> calc
irrational -> complex, non-algebraic, non-solvable, non-computable: more
abstract and generic
later %%%%%%%%%%%%%%
later issues, like solving polynomials, now well-understood;
. not all questions these provoke are solved (absolute galois group of
Q)
. some elementary conjectures still open (Goldbach/Goldberg sp?; Fermat
until recently)
Other than these elementary conjectures,
math is mostly working on the things discovered in the 19th century.
(rigor and foundations of old things have been studied;
not "resolved", but well-developed)
Also, new symmetries (finite simple groups, Lie algebras) discovered,
in the line of Platonic solids.
(indeed, Coxeter clarified Platonic solids!)
Pithily:
- Everything the Greeks asked has been answered
(in the 19th century),
- but these answers provoked new questions, which are still open