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# Organizing principles of mathematicmathematics

In his famous paper "The two cultures of Mathematics"(http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf) T.Gowers 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 coecients 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 defnitions 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 varyous various areas? I mean, for example, globalization technic 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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# Organizing principles of mathematic

In his famous paper "The two cultures of Mathematics" (http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf) 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 coecients 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 defnitions 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 varyous areas? I mean, for example, globalization technic in topology (structure functor Hirsh, Differential Topology, $\S 2.11$ and Mayer–Vietoris sequence, 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.