- motivating theories
+ nice language
+ application
+ natural generalization
(modules as analog over ring; R^p+q as generalization of Euclidean)
[or Lie groups as "continuous groups"]
+ internal beauty
(Lie theory)
Some folk *like* heavy duty;
some really don't.
[Beware of formal motivations:
- should at least get better perspective on something you already know
- ...but much better to know some original context
]
You can't win over skeptics.
(or it's very hard)
heck, how do you justify *primes*?
largely: internal beauty of number theory
[or application to cryptography]
eg linear algebra:
matrices sound like "just a language" for solving systems of equations
eg of Gauss-Jordan
ok, it's nice that
OTOH, closed form expression for Fibonacci??
what about modules?
Jordan form of matrices really needs modules
(can do with "generalized eigenspaces" only etc.,
but that's clunky: you're doing module theory in disguise)
algebraic varieties (via Spec), say to define intersection number algebraically
"just a language"
Galois theory is an application of group theory to natural question:
not "just a language"
rude: "it's just a language"
polite: "It's a very elegant, unifying language. What new does it prove?"
[Category theory is weird in that motivating examples were originally
in computing the homology of a somewhat pathological space,
the solonoid]
motivation for category theory / adjoint:
Grothendieck used category theory as a unifying language
(in functional analysis, homological algebra and algebraic geometry)
He proved relative Serre duality,
which used adjointness fundamentally.