Mathematics has this funny habit of throwing around "it is easy to say that ...", or in this case, calling a proof that eluded mathematicians for decades "simple."

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"

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.

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

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.

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

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.

Simple (easy to say) is not the same as being easy to deduce. A needle is a simple object, but finding one in a pile of hay takes a lot of work. Likewise, it is hard to pick out the features that make a problem simple when they are embedded in a sea of complexity. It becomes easy to say once you are no longer faced with the difficulty of knowing the right thing to say.

>Simple to say is not simple to deduce.

Case in point: Fermat's Last Theorem, or Collatz Conjecture

Well, no, the point isn't that simple propositions may have complex proofs, the point is that a proof may itself be simple, though hard to discover in the first place.

Alternatively I could step on the hay with a bare foot, because if the needle is in there, I'll step on it for sure.

Or you could just burn the hay.

Easy to prove =/= Easy to discover (+).

It is simple to say E=MC^2.

That doesn't take away from the fact that it was very hard for it to be discovered.

And it's a good thing. It's a good thing that things which were hard to discover can be communicated easily. Otherwise we wouldn't be able to compress thousands of lifetimes worth of discoveries into a one-semester lecture.

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(+) footnote: well, unless P=NP.

Abstract link: https://arxiv.org/abs/1408.1028