Overview of math
- several sources
(counting, measuring, logic:
arthimetic, geometry, logic: 3 liberal arts (2 from Quad, logic from Trium))
(note that music & astronomy were also very mathy)
other 2 were Grammar & Rhetoric: language)
Euduxous & Archimedes -> Calc
(just with 2000 years in between of stagnation)
- 19th century
2 big questions:
- solving polynomials
- solving differential equations
Turns out many are *unsolvable*,
and that solvability is deeply related to the *symmetries* and
*geometry* of the problem.
Polynomials involve discrete symmetries/geometry (groups),
while differential equations involve continuous symmetries/geometry
(Lie groups).
Summarized in Klein's great geometrization/unification.
(geometry & symmetry)
(His "Erlangen" program;
see:
http://math.ucr.edu/home/baez/erlangen/
...for details
)
Understanding the structures discovered in the 19th century is still
occupying mathematicians:
19th century was great flowering.
[This is why Gauss is so central.
Yes, he was brilliant, but he came at a good time:
we crested the greatest hill / found richest valley.]
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Mainstream?
(algebraic geometry? Lie theory?)
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Why is all math connected?
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New fields
Study pieces that develop,
and sometimes very deep.
(topology, for instance)
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Aside on "math/science and the times":
people say that Goedel's incompleteness was totally 20th century
"limits of logic/knowledge" etc.,
but what are we to make of Abel/Galois showing that quintics aren't solvable?
Does *that* fit into some sorta Victoria world?
Time of revolution?
No. It just doesn't fit the narrative, so it's ignored.
19th/20th century
also had some clean-up
e.g., Hilbert's *correction* of Euclid!
(needed order axioms)
(Note that this was a tour de force, but *sterile*)
...some of which lead to deep results.