There is a core of structured topics, and many auxiliary ones
rigor is a deep theme, in two bursts: Greeks and 19th century
[paradigm shifts]
. basics many places
. Greeks had proofs
. dark ages
(some science, very little math)
. revival around calculus and polynomials
(and some number theory)
. led to 19th century: Galois, Lie, etc.
also more rigor (like Greeks, rigor reveals deeper structure)
. 20th century
lots of time understanding central objects more deeply
(connections, categories, homological algebra, finite simple groups,
Lie theory, etc.)
floodgates: lotta new fields (still arising): set theory, information,
error correcting codes, coarse geometry, etc., etc.;
(lack of unity: Leibniz last to know everything in West;
Gauss last one to know everything in math; Hilbert last universal
mathematician (worked in all fields))
Math very part of modernity:
post-19th century math feels v. far removed from classical math
[like Miro or abstract expressionists do from 19th century painting]
There *are* connections,
and it did develop organically,
but it was a big paradigm shift:
college math feels like an alien planet after calc class.