[FOM] Dirichlet's theorem; boiling down proofs
Timothy Y. Chow
tchow at alum.mit.edu
Sat Aug 18 22:26:39 EDT 2007
Joe Shipman wrote:
>I propose the thesis "any mathematics result more than a century old is
>suitable for undergraduate math majors".
>Another version of the thesis is "any mathematics result more than 200
>years old is suitable for freshmen"
A provocative thesis, but surely untenable. You're probably forgetting
the enormous amount of sophisticated work in what we would today call
"analysis and applied mathematics." There's a ton of 19th century work on
special functions (elliptic modular functions, abelian functions, theta
functions, fuchsian functions...) that is way beyond today's
undergraduates. Ditto with Lie theory, both in the abstract and as
applied to differential equations. The unsolvability of the three-body
problem by first integrals would also seem too advanced.
In number theory, the Kronecker-Weber theorem is probably even more
challenging than Dirichlet's theorem.
Also, what about things like Riemann's existence theorem, or the Schubert
calculus and other results in algebraic geometry? Do these not count,
because by modern standards of rigor they weren't "cleaned up" fully until
the 20th century?
As for the 200-year mark, one touchstone is Gauss's Disquisitiones
Arithmeticae (1801). Much of this material is taught to Math Olympiad
types in high school, but I don't know that it's suitable for the typical
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