Joe Shipman recently posted an interesting e-mail on the Foundations of Math e-mail list:
I propose the thesis “any mathematics result more than a century old is suitable for undergraduate math majors”.
Note that the original proofs may be too difficult for undergraduates, I am only requiring that today a “boiled-down” proof (which may be embedded in a much larger theory than existed at the time of the original proof) could be taught.
So far I have only found one significant counterexample, Dirichlet’s theorem (which, in its logically simplest form, states that if a is prime to b, there exists a prime congruent to a mod b).
Can anyone think of better counterexamples? Does anyone know of a proof of Dirichlet’s theorem that does not require prerequisites beyond the standard undergraduate curriculum?
(Two other possible counterexamples, the Prime Number Theorem and the Transcendence of Pi, are proven sufficiently easily at the following links that they would, in my opinion, be appropriate for a senior seminar:
Another version of the thesis is “any mathematics result more than 200 years old is suitable for freshmen” (note that most high schools offer a full year of Calculus). Results that were merely conjectured more than 200 years ago but not really proved until later don’t count.
I’ve sometimes considered something like this. Can anyone else think of potential counterexamples? I wonder if there were some results known on solutions of differential equations in the 18th century that would be too advanced for first-years. And probably some particular calculations done in the 19th century that are just too large for an undergraduate to properly manage. I think it’s also possible that some of Cantor’s results on the possible Borel structures of the sets of discontinuities of real-valued functions might be too advanced, but it’s also possible that advanced seniors can manage them. Or perhaps the Riemann-Roch theorem? (I don’t actually even know enough to state that theorem myself.)
Another interesting corollary to this discussion – what’s the earliest result of the 20th century that is beyond the reach of an advanced undergraduate?