In his famous paper "The two cultures of Mathematics" T. Gowers gives examples of organizing principles in combinatorics.

(i) Obviously if events $E1, \cdots,E_n$ are independent and have non-zero probability, then with non-zero probability they all happen at once. In fact, this can be usefully true even if there is a very limited dependence. [EL,J]

(ii) All graphs are basically made out of a few random-like pieces, and we know how those behave. [Sze]

(iii) If one is counting solutions, inside a given set, to a linear equation, then it is enough, and usually easier, to estimate Fourier coefficients of the characteristic function of the set.

(iv) Many of the properties associated with random graphs are equivalent, and can therefore be taken as sensible definitions of pseudo-random graphs. [CGW,T]

(v) Sometimes, the set of all eventually zero sequences of zeros and ones is a good model for separable Banach spaces, or at least allows one to generate interesting hypotheses.

(vi) Concentration of Measure

More examples (by Tao and other) you can see at http://ncatlab.org/davidcorfield/show/Two+Cultures

Do you know another examples in various areas? I mean, for example, globalization techniques in topology (structure functor in Hirsh, Differential Topology, $\S 2.11$ and Mayer–Vietoris sequence, in Bott & Tu, Differential Forms in Algebraic Topology $\S 5$).

So, many proofs look like "prove the local version of theorem and globalize".

Do you know such principles? It should be more specific than undergraduate course but it should be common used in your branch and be situated in "common wisdom" of mathematics.

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    $\begingroup$ community wiki? $\endgroup$ – Ed Dean Dec 8 '10 at 23:56
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    $\begingroup$ Somewhat related questions: mathoverflow.net/questions/1890/… and mathoverflow.net/questions/8874/… $\endgroup$ – Gerry Myerson Dec 9 '10 at 0:19
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    $\begingroup$ Seriously, these are not organized principles of combinatorics. Asymptotic combinatorics, maybe. $\endgroup$ – darij grinberg Dec 9 '10 at 9:50
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    $\begingroup$ My precise words, for what it's worth: "My main point about such principles is not so much that they are useful, which is not particularly surprising, but that they play the organizing role in combinatorics that deep theorems of great generality play in more theoretical subjects." And earlier: "I often use the word 'combinatorics' not quite in its conventional sense, but as a general term to refer to problems that it is reasonable to attack more or less from first principles. $\endgroup$ – gowers Dec 9 '10 at 11:59
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    $\begingroup$ My point is that all of your principles are from probabilistic, asymptotic, in-the-long-run combinatorics, and usually give bad bounds - "bad" not in the sense of "useless" or "bad mathematics", but in the sense of "there is much space between the bound and the actual values" and in the sense of "many constants appear that are almost impossible to calculate explicitly". On the other hand, a lot of combinatorics is about very precise and sharp bounds. ... $\endgroup$ – darij grinberg Dec 12 '10 at 0:07

The Choquet theory in convex analysis / functional analysis / whatever you want to call it. An element of a convex set should be some kind of "average" of extreme points. This has the status of a theorem for compact sets in normed linear spaces but is a useful guiding principle for not-necessarily-compact sets in not-necessarily-normed linear spaces. Chapter 14 in Lax's Functional Analysis book gives good examples of the wide array of applications of the same simple idea.


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