3 ms·

You just need to understand the language. "Trivial" = "regularly covered in undergraduate courses" "Easy" = "a good topic for an undergraduate honour's thesis

You just need to understand the language. "Trivial" = "regularly covered in undergraduate courses" "Easy" = "a good topic for an undergraduate honour's thesis" "Non-trivial" = "a good PhD thesis topic" "Distinctly non-trivial" = "a groundbreaking result which will establish a professor's reputation in the field"

tnecniv 10y ago
I had a professor who liked to talk about how "elementary" did not mean easy, it just means "uses only the foundations." This proof is a perfect example of that statement. Formulating the problem the right way -- which is most often the hardest part -- was the real challenge here, not the mechanisms needed to do the formulation or the proof.
- huhtenberg 10y ago
  Not sure why, but this reminds me how we had a professor that would say Let y(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e, whereby e is not _necessarily_ the base of natural logarithm
  - cperciva 10y ago
    My favourite along those lines was "let epsilon be a small number which is not necessarily greater than zero". Everybody who spent the preceding year on epsilon-delta proofs did a double-take at that.
- FabHK 10y ago
  There's a wonderful book, by the way, Proofs from THE BOOK, of, well, simple beautiful proofs. The book is named after > mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book." https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
sacheendra 10y ago
This made my day!
MaysonL 10y ago
And of course, those change over time. I had an undergraduate course my freshman year where we went through a circular proof of the equivalence of twelve or thirteen formulations of the axiom of choice. A hundred years ago, proving many of the steps of that proof might well have been non-trivial, perhaps even distinctly so.

tnecniv 10y ago

I had a professor who liked to talk about how "elementary" did not mean easy, it just means "uses only the foundations." This proof is a perfect example of that statement. Formulating the problem the right way -- which is most often the hardest part -- was the real challenge here, not the mechanisms needed to do the formulation or the proof.

huhtenberg 10y ago
Not sure why, but this reminds me how we had a professor that would say Let y(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e, whereby e is not _necessarily_ the base of natural logarithm
- cperciva 10y ago
  My favourite along those lines was "let epsilon be a small number which is not necessarily greater than zero". Everybody who spent the preceding year on epsilon-delta proofs did a double-take at that.
FabHK 10y ago
There's a wonderful book, by the way, Proofs from THE BOOK, of, well, simple beautiful proofs. The book is named after > mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book." https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

sacheendra 10y ago

This made my day!

MaysonL 10y ago

And of course, those change over time. I had an undergraduate course my freshman year where we went through a circular proof of the equivalence of twelve or thirteen formulations of the axiom of choice. A hundred years ago, proving many of the steps of that proof might well have been non-trivial, perhaps even distinctly so.