What's the most useful piece of mathematical "folk wisdom" you've encountered? I'm talking here about things that aren't theorems, or even conjectures, or even shadows of conjectures -- just broad analogies or slogans that nonetheless tend to be incredibly useful for formulating conjectures, understanding when a line of attack on a problem will or won't work, etc.

The first example that leaps to my mind is the Cramer probabilistic model of the primes, which tends to give astoundingly good predictions for all sorts of things despite being horribly simple.

Oh, right: Community wiki, one tidbit of wisdom per post, please!

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I would also add the "intuition" tag to this question. –  Jose Brox Nov 12 '09 at 17:21
I appreciate posts of this type, but I expect that a more focused request for a particular class of heuristics would be more successful. The Tricki's categories could serve as a guide here. For instance, one could make a request for "Techniques for finding algorithms and algorithmic proofs". –  Jack Lemon Dec 7 '10 at 0:00

Rather than suggest one heuristic, I thought I would point out that the Tricki (http://www.tricki.org) is a repository of useful heuristics.

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"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard

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Putting it on here to be up/downvoted with the rest: Cramer's probabilistic model of the primes, wherein an integer n is prime with probability 1/log n.

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For psychological reasons, though, the Cramer model really offends some mathematicians who aren't used to thinking probabilistically. They inevitably ask "but what do you mean, a number has a probability of being prime?" –  Michael Lugo Oct 28 '09 at 16:18

I like Littlewood's three principles of real analysis

1. Every measurable set is nearly a finite sum of intervals

2. Every L_p function is nearly continuous

3. Every convergent sequence of functions is nearly uniformly convergent,

of which 2 and 3 are in fact Lusin's and Egorov's Theorems.

http://en.wikipedia.org/wiki/Littlewood%27s_three_principles_of_real_analysis

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A very interesting heuristic is a principle in complex variables called "Bloch's heuristic principle".

Bloch's principle is about families of analytic functions called "normal". A family F of analytic functions on a domain D is called normal on D if every sequence of functions of F has a subsequence that converges uniformly on compact subsets (either to an analytic function or to infinity). Normal families are very studied for their applications in complex dynamics.

Bloch's principle goes as follows :

A family of analytic functions on a domain D having a property in common is most likely to be normal if there is no non-constant entire function having this property on the whole complex plane.

There are many examples of Bloch's principle. For example, take the property of being bounded : a well known theorem of Montel says that a family of analytic functions on a domain D which is uniformly bounded is necessarily normal on D, and Liouville's theorem says that there is no non-constant entire bounded function.

Or, take the property of omitting two distinct complex values. Again, a theorem of Montel says that a family of analytic functions on a domain D such that each function omits a,b in C, a different than b, is normal on D. The version for the whole complex plane is a well known theorem of Picard, that says that there is no non-constant entire function that omits two distinct complex values.

However, there are many counter-examples to Bloch's principle as it is stated, but it can be transformed into a rigourous theorem that goes like "If a property satisfies these conditions, then bloch's principle is respected".

I wouldn't qualify Bloch's principle as "most helpful", but it is certainly interesting.

Malik

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I don't know how useful it is in front-line research, but one of my former probability lecturers once made the following suggestion: "When in doubt, interchange the order of integration and summation. Miracles can occur."

On a more temperate note, although Fubini's theorem isn't a heuristic, when correctly applied, it's often worth ignoring integrability/convergence/a.e. issues and doing some formal manipulations, as long as one then goes through it again properly.

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Any identity of power series that holds for sufficiently small values of the variable(s) also holds for formal power series.

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This can be turned into a theorem, though. –  Qiaochu Yuan Oct 29 '09 at 6:33

Dear harrison,

in complex geometry there is Oka's principle. It says that on a stein manifold, if there is no obstruction to a continuous construction there will be no obstruction to a holomorphic construction either.

[A stein manifold is a holomorphic submanifold of C^n : it is the analogue in the complex analytic category of affine varieties.]

The simplest example: consider an open subset U of C and a point p in U. Given a holomorphic function on U-p, it can be holomorphically extended across p as soon as it can be continuously extended ( U is stein, as is every connected non-compact complex manifold of dimension one: Behnke-Stein's theorem).

Next simplest example: on a stein manifold two holomorphic line bundles are analytically isomorphic as soon as they are topologically isomorphic.

This last result was vastly generalized by Grauert to the case of an arbitrary principal bundle( with group an arbitrary complex lie group) over a stein manifold: a remarkable achievement.

And Grauert's theorem has been generalized by Gromov to a new instance of Oka's principle...

You will find a very poetic homage to Oka and his principle in the following article

http://www.jstage.jst.go.jp/article/kyushumfs/33/1/83/_pdf

in which J.Kajiwara explains that Oka lives in Buddha's paradise.

Friendly, Georges.

PS Here is a very exhaustive survey by Pit-Mann Wong I have just found by googling

http://www.nd.edu/~pmwong/OKA.pdf

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A sort-of heuristic in combinatorics is that if you can't figure out what to do with a set, take the free abelian group / vector space on that set and work with linear transformations instead of functions. For example, there is a certain condition on certain linear operators one can define on the free vector space on a graded poset that guarantees the ranks are unimodal, and for many applications of this condition a direct proof of unimodality isn't known.

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I don't remember where I heard this, but I recall hearing about a mathematician who said that if an assertion was true for a general 3x3 matrix, then he would believe that same assertion with confidence for any square matrix.

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Here's a related quote: "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." -- Olga Tausky-Todd –  John D. Cook Oct 29 '09 at 2:44
Ah, but I think the 2x2 and 3x3 cases are really different! There is a lot of instances when GL_2(A) or SL_2(A) (groups of 2x2 matrices over a ring A) exhibit a behavior different from the general nxn case. Also, sl(2) stands out, in many respects, of other simple classical Lie algebras. –  Pasha Zusmanovich Nov 3 '09 at 12:07

It's sometimes useful to think of log x as (x^0 / 0), which is what you get by integrating x^(-1) using the usual formula for integrating x^n when n is not -1.

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Weinstein's Lagrangian creed: Everything is a Lagrangian submanifold

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Every set or function you can build in a concrete fashion starting from (countablye many) other measurable sets or functions is measurable.

There are some counterexample to this. But it gets you a feel for measurability like we have a feel for what is continuous and what is not.

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In finite group theory: if your conjecture is true for {2,3} groups it is probably true for all soluble groups.

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Three of the most useful heuristic rules I've learnt:

(1) When trying to define a morphism that has to make a diagram commutative, first forget about signs: in the end, they will agree.

(2) If you want to prove something using induction, first check if it works (besides for n=1) for n=2.

(3) Trying to define a morphism that has to fulfil some universal property? -The first one (diferent from zero or constant) that you can imagine / find is probably: (a) the only non trivial one that exists, (b) that miraculously fulfils also your requirements.

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I'm not sure those are very useful. There are many cases where all fail, too many for the heuristic to be truly useful in my opinion. The first example that came to mind was a particular proof of Poincaré Duality where the diagram only turns out to be sign-commutative; examples for the other cases are quite easy to come up with too. –  Sam Derbyshire Oct 29 '09 at 1:09