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Results tagged with big-list
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user 1459
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
18
votes
Slick ways to make annoying verifications
The probabilistic method is a good source of slick proofs of things that are either very hard to prove or even not known to be provable in any other way. For example, suppose you are asked to prove th …
7
votes
Slick ways to make annoying verifications
Sometimes in elementary analysis there are things that are a pain to check, but one can at least minimize the pain. For example, if you want to prove that for every $\delta > 0$ the sequence $(1+\delt …
50
votes
Ingenuity in mathematics
The impossibility of tiling a chessboard with two opposite corners removed using dominos is quite good for this purpose I think, especially if you start by giving a boring case-analysis proof for a 4- …
7
votes
Examples of statements that provably can't be proved using a promising looking method
An elementary example is the fact that unique factorization fails in $Z[\sqrt{-5}]$, which shows that there is no obvious proof of the fundamental theorem of arithmetic.
8
votes
Examples of statements that provably can't be proved using a promising looking method
Another elementary example concerns the fact that the integers are unbounded in R. It surprises many people that the completeness axiom is needed to prove that, but one can show that that is the case …
14
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
I don't know whether you would count this, but the proof of the existence of quotient groups seems to fit your description. Some people define the product of the cosets $gH$ and $g'H$ to be the coset …
8
votes
Casual tours around proofs
Timothy Chow's article on forcing (called A Beginner's Guide to Forcing) is one of the best of this general type.
http://www-math.mit.edu/~tchow/forcing.pdf
15
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
Another thought is the path-integral formulation of quantum mechanics, where one integrates over all possible histories of a system, with appropriate weights.
32
votes
Situations where “naturally occurring” mathematical objects behave very differently from “ty...
Thanks to Boris Tsirelson, we know that not all infinite-dimensional Banach spaces contain either $c_0$ or $\ell_p$ for some $p\in [1,\infty)$. But all known counterexamples are constructed in a parti …
34
votes
Accepted
Is there any geometry where the triangle inequality fails?
There are people who seriously study quasi-normed spaces. The most natural examples are $\ell_p$ spaces for p strictly between 0 and 1 (the "norm" given by the usual formula and the distance given by …
- 29k
7
votes
Papers that debunk common myths in the history of mathematics
Historians of mathematics get very cross if you dare to say "Pythagoras's theorem." You have to say, "The Pythagorean theorem," to emphasize that it wasn't the brainwave of a man called Pythagoras. In …
117
votes
Books you would like to read (if somebody would just write them…)
I don't know for certain that this doesn't exist, so I'm in a no-lose situation: if this is a rubbish answer then it means that a book that I want to exist does exist. Many mathematicians of a pure be …
30
votes
Generalizing a problem to make it easier
I can't resist mentioning the following problem (and requesting that nobody gives away the solution any more than it is already given away by my mentioning it here).
Call a real number repetitive if …
20
votes
Different ways of proving that two sets are equal
In a similar spirit to the dense+closed answer, there are some proofs where to show that a subset of a connected space is the whole space, one shows that it is non-empty, open and closed. An example o …
10
votes
Ways to prove the fundamental theorem of algebra
Here's another complex analysis proof that I heard about for the first time under a week ago (because it was set as a question on a course I am teaching for). Pick a circle large enough for the modulu …