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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.
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 …
- 29k
15
votes
Examples of theorems misapplied to non-mathematical contexts
This is a wonderful and fascinating still life by Juan Sanchez Cotán: https://www.khanacademy.org/humanities/monarchy-enlightenment/baroque-art1/spain/a/juan-sanchez-de-cotn-quince-melon-and-cucumber
…
- 6,237
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 …
- 12.9k
73
votes
What are some reasonable-sounding statements that are independent of ZFC?
My favourite is the statement that if $X$ is a set of reals, and for every sequence $(a_n)$ of positive reals you can find a sequence of intervals $(I_n)$ that cover $X$ such that $I_n$ has length at …
- 162
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 …
- 101
37
votes
Experimental mathematics leading to major advances
Another thing you might like to check out is Herb Wilf's very nice article: Mathematics, an experimental science in the Princeton Companion to Mathematics, in which he talks about the interplay betwee …
- 24.7k
20
votes
What are some fundamental "sources" for the appearance of pi in mathematics?
Let me play devil's advocate here: I'm not sure that I agree that the ubiquity of π is so mysterious. After all, how do you ever prove that π appears? You have to relate your situation to some known s …
- 456
58
votes
Intuitive crutches for higher dimensional thinking
This is a slightly different point, but Vitali Milman, who works in high-dimensional convexity, likes to draw high-dimensional convex bodies in a non-convex way. This is to convey the point that if yo …
- 24.7k
21
votes
Are there any good websites for hosting discussions of mathematical papers?
Time to mention the Selected Papers Network: https://selectedpapers.net
EDIT:
I hope that Tim does not mind me editing this to add extra links:
Introductory discussion by John Baez: http://johncar …
- 56.9k
13
votes
Believing the Conjectures
I initially wrote this as a comment, but it got too long and it sort of contains an example, so here goes. Reflection seems false in a number of contexts, since there are many properties that can't be …
- 29k
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
- 29k
8
votes
What are some slogans that express mathematical tricks?
Pick a random example.
If you add lots of small and reasonably independent things together then the result will be highly concentrated about its mean.
- 29k
2
votes
Video lectures of mathematics courses available online for free
Eight recent lectures by Emmanuel Candes on compressed sensing are linked to from here: http://www.newton.ac.uk/programmes/INI/iniw04p.html
More generally, the Newton Institute has been making a larg …
- 29k
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 …
- 29k
49
votes
Theorems that are 'obvious' but hard to prove
There are a number of facts in multivariable calculus that are obvious but hard to prove. For instance, the change-of-variables formula in a multiple integral is very easy to justify heuristically by …
- 29k