4 ms·
Not really a proof, just stating the crux of the proof. If this is considered a proof then any proof can be converted to one sentence. For example, proof that
by Grue3 10y ago
Not really a proof, just stating the crux of the proof. If this is considered a proof then any proof can be converted to one sentence.
For example, proof that real numbers are not countable:
If r_n is the n-th real number in [0,1) and n{k} is kth digit of n in base 3, then consider sum(3 - r_j{j})/3^j (QED)
- tgb 10y agoI'm a grad student in math and I can say that sentences like this are quite typical in published proofs. Details are often omitted (though I just found a mistake in a paper hidden in a 'left to the reader' statement).
- RossBencina 10y agoObligatory links to Lamport's "How to Write a Proof" [0], and "How to Write a 21st Century Proof."[1] A video presentation of the latter is discussed at [2]. [0] http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf http://research.microsoft.com/en-us/um/people/lamport/pubs/l... [1] http://research.microsoft.com/en-us/um/people/lamport/pubs/proof.pdf http://research.microsoft.com/en-us/um/people/lamport/pubs/p... [2] https://news.ycombinator.com/item?id=10689391 https://news.ycombinator.com/item?id=10689391 [edit: formatting]
- EGreg 10y agoI thought unprovable statements are usually the funniest to "leave to the reader". Like that time someone solved an unsolved problem or two because it was on his homework: https://en.m.wikipedia.org/wiki/George_Dantzig https://en.m.wikipedia.org/wiki/George_Dantzig
- fovc 10y agoI can't seem to find it, but I was under the impression that some famous mathematician (I had Erdos in mind, but that's clearly wrong) solved an open problem during a math olympiad in the 1950s. Perhaps that was just an tall tale our trainers would tell us, though!
- pmiller2 10y agoErdős would have been nearly 40 in 1950, FYI (far too old to be competing in math olympiads).
- nightcracker 10y agoThe best method for proving theorems is proof by intimidation.
- kmill 10y agoSpeaking of uncountability, here's a similar one for why the set of functions N->{0,1} is not countable. If f:(N->{0,1})->N were a bijection, let g(t)=1-f^(-1)(t)(t), and compute g(f(g)). Anyway, whether something counts as a proof is a matter of whether you (logically) accept it as irrefutable evidence. As with any genre, you have to consider your audience. I hear gauge theorists are content to have as a proof a couple of references to a couple papers which have techniques which apply, with minor alteration, to the current problem. That said, I wouldn't have minded if Zagier actually demonstrated it was an involution with exactly one fixed point!