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Today's xkcd describes magic proofs. Can anyone think of some good examples of these?

kdog
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    It is generally considered to be a non-proof. Here is a list of other invalid proof techniques. – JMoravitz Aug 25 '16 at 20:32
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    @JMoravitz That depends on what exactly "it" is. – Robert Israel Aug 25 '16 at 20:36
  • I would say there kind of is such a thing but it's not very well-defined and (as is often the case) kind of more to do with not understanding what's really going on/seeing something for the first time than anything deeper. – Thompson Aug 25 '16 at 20:37
  • Zagier's one sentence proof of Fermat's theorem on the sums of two squares feels like dark magic to me. (But no guarantees on whether other people find it to be dark magic or just obtuse) – Milo Brandt Aug 25 '16 at 20:41
  • I understand what you mean, kdog, but I think what feels like "dark magic" varies heavily from person to person, and depends on where you are in your math education. The first time I saw a proof of irrationality of $\sqrt{2}$ it felt like magic, because I had never seen an argument go like that. But now it feels like a standard way of determining rationality. That said, you might enjoy http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice – Alex Meiburg Aug 25 '16 at 21:06
  • The Infinite Improbability Drive? – Calum Gilhooley Aug 25 '16 at 21:26
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    I voted to close this is opinion based, I might also have chose too broad or unclear (which were reasons selected by others). The point is that it is not precise what is intended there. For example I do not think that @JMoravitz examples are examples of what is meant there, but maybe they are. Who knows for certain. I also do not think it is generally about proofs by contradiction, or surprising "magic" proofs. – quid Aug 25 '16 at 21:38
  • I read that kdog wants some more math 'funnies'. There was one on the math-room wall with some ancient Prof. doing a calculation on a chalk board with the words "..and then a miracle occurs.." in the middle of his 'proof'. – Daniel Buck Aug 26 '16 at 20:02

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explainxkcd has some great explanations for those questions. Just add "explain" in the url:

http://xkcd.com/1724/ becomes http://explainxkcd.com/1724/

For example, in 1931 Kurt Gödel was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the set theory, where many statements become impossible to prove and disprove if the axiom of choice is not taken as a part of the axiomatic system.

Kaligule
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