Here's an ultra-brief thumbnail of history of mathematics:

Why this love of the -Greeks-?
B/c they had -reason- and -rigor- in a way that AFAIK no-one else in
the world did -- and these are the -basis- of math.
(others did good math, but didn't go as far as the Greeks b/c you
 need -rigor- to build a good foundation; EG, Egyptians didn't -prove-
 anything: they just stated it, from authority. I suspect this
 happened in China too...
 This is likely related to democratic ideals.)

Greeks:
- did geometry;
  their numbers were all (?) expressed as ratios of lengths/areas/volumes
  (you didn't square a number: you constructed a square with that
  -length- as side -- lengths and areas were incomensurable)
  Archimedes: rigor, infinity
    Archimedes formulated our intuitions of -rigor-:
    it was not enough to argue plausibly, rhetorically.
    He also formalized the notion of "arbitrarily large"
  Euclid: axioms
    (that is, lots of stuff was known -- but Euclid wrote it up
     as proceeding from -axioms-)
  Pythagorus: discovers irrational numbers
    (that's why he was more valuable than most other peoples
     who proved the pythagorean theorem but didn't discover
     irrationality)
  platonic solids:
    tetrahedron, cube/octahedron, dodecahedron/icosahedron
  number theory:
    rationality/irrationality, factorization, prime numbers
  calculus:
    the method of -exhaustion- (needed to compute volume of cone and area
    of parabola) and Archimedes invention of Cavalieri's principle (to
    compute volume of sphere)
  limitation:
    they only worked with lines and circles, later with conics:
    but still only -quadratics-.
    Babylonians actually had smooth cubics!!
    Which are -very- interesting!!!

Big unsolved problems:
 "3 problems of antiquity":
 - squaring circle (/rounding square)
 - doubling cube
 - trisect an angle
Niggle:
 parallel line postulate (when did this start niggling?)

...some later greeks looked at things like the nodal cubic curve
($y^2=x^3$, for doubling the cube)

China/India/Arabs
  (NB: while today's chinese mathematicians are brilliant (no racist
  bullshit, please), chinese mathematics knew little compared with the
  Greeks: EG, couldn't compute volume of cones!)
some advances:
  - decimal number system (EG chinese)
  - zero (desi/arab thing: in china just left a space)
  others? (did good science...)

  descartes, analytic geometry?
  (huge deal. huge deal.)
  (in particular, relates euclidean geometry to -equations-:
   connects algebra and geometry)
17th century :: calculus
  started with Galileo, ended with Newton?/Bernoulli?
  birth of modern physics, deeply tied to mathematics
18th century
  ???
  Euler?
  - complex numbers/fundamental theorem of algebra??
  (solution of cubic/quartic?)
19th century
 - rigor: deep increase in rigor:
   making calculus rigorous lead to advances in Real Analysis
 - Galois theory/field theory
   ...including "no general solution of quintic"
   ...leads to deep advances in number theory
 - group theory
 - solutions (via field theory) of "3 great problems of antiquity"
   ...including "pi is transcendental"
 - non-Euclidean geometry: resolution of "parallel line postulate"
   (namely, you -can- have more geometries...let's study 'em!)
 - Riemannian manifolds
 - linear algebra and higher dimensions
 - Lie groups/algebras
 - Cantor and set theory
 - construction of the real numbers
 - algebraic geometry? (when did the Italians get their start?)
 - Klein's program of "geometric structures as something preserved by
   a group"

20th century
 - huge amount of activity: no longer possible to keep up
 - Hilbert's problems: an ecletic set, but motivated -some- good
   mathematics
 - cones need calculus to compute volume (cute)
 - Godel/Cohen: incompleteness, independence
 - algebraic topology: algebraic invariants of geometric spaces
 - category theory: a huge unification
 - computers: mathematics key in underpinings of computers
   (note von Neumann's great insight: programs and data aren't
   separate! by making programs just a kind of data, he allowed
   introspection and made computers incredible. Also lead to the
   security problems intrinsic to that: you can't separate data from
   commands)
 - cryptography: huge advances in secret communications
   (including public key, which makes the secure internet possible)
 - Coxeter's detailed study of regular solids and connections
   to representation theory
 - information theory, complexity, crazy dynamical systems,
   instability, etc.
 - arithmetic algebraic geometry:
   much number theory becomes a rigid kind of -geometry-

21st
 - math is huge -- this is the golden age
