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Many "tricks" that we use to solve mathematical problems don't correspond nicely to theorems or lemmas, or anything close to that rigorous. Instead they take the form of analogies, or general methods of proof more specialized than "induction" or "reductio ad absurdum" but applicable to a number of problems. These can often be summed up in a "slogan" of a couple of sentences or less, that's not fully precise but still manages to convey information. What are some of your favorite such tricks, expressed as slogans?

(Note: By "slogan" I don't necessarily mean that it has to be a well-known statement, like Hadamard's "the shortest path..." quote. Just that it's fairly short and reasonably catchy.)

Justifying blather: Yes, I'm aware of the Tricki, but I still think this is a useful question for the following reasons:

Right now, MO is considerably more active than the Tricki, which still posts new articles occasionally but not at anything like the rate at which people contribute to MO.
Perhaps causally related to (1), writing a Tricki article requires a fairly solid investment of time and effort. The point of slogans is that they can be communicated without much of either. If you want, you can think of this question as "Possible titles for Tricki articles," although that's by no means its only or even main purpose.

edited Dec 14 at 18:51

Anton Geraschenko♦♦

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4 revisions, 2 users
Harrison Brown 100%

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I realize this question's borderline (and I wasn't sure whether I should ask it), but I don't think it's any less MO-appropriate than "Fundamental examples" or the mathematical joke thread. People who downvoted, care to explain why you disagree? – Harrison Brown Dec 14 at 15:31

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I wasn't the downvoter, but I think some people are getting a little annoyed at the number of these "produce a ginormous list of answers" soft questions. You're entirely right that there are worse offenders on the site, but I think the issue is in part the density of them, not any particular ones. – Ben Webster♦ Dec 14 at 16:07

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@Ben: That makes sense, although a better solution might be to ignore the "soft-question" tag. Or make a new tag for "ginormous list" questions. But this would be more appropriate on meta, so I'll start a topic there. – Harrison Brown Dec 14 at 16:11

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Ben: I really like the ginormous list of answer questions. It's nice to have some big picture responses from a variety of mathematicians. Peter: If you don't like soft answer questions, you don't have to read them. There's even a box on the front page for ignoring tags you don't like. – Tom LaGatta Jan 13 at 9:51

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@Tom: soft-answer is not the same as big-list; the meta discussion that came from this thread actually inspired the "big-list" tag, for exactly the reason that it can be ignored. But if you have more to say, feel free to contribute on meta.MO. – Harrison Brown Jan 13 at 11:01

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Darsh Ranjan · Answer 1 · 2009-12-15 01:40:43Z UTC

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The analyst's toolbox consists of three things:

The Cauchy-Schwarz inequality
Changing the order of integration/summation
Integration by parts

(I'm not saying I believe that; it's just a very common saying.)

answered Dec 15 at 1:40

Darsh Ranjan

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You forgot 4. Adding and substracting something. – Mariano Suárez-Alvarez Jan 13 at 7:37

"The Cauchy-Schwarz Master Class" by J. Michael Steele shows how you can do a surprising amount with just the first item on your list, and even more with occasional help from the other two items. – John D. Cook Jan 13 at 13:08

shenghao · Answer 2 · 2009-12-15 04:15:40Z UTC

Try to replace a structure on an object with a map to a clasifying object.

E.g., replace a cohomology class of a space with a map to an Eilenberg-MacLane space. Replace a vector/general bundle on a manifold with a map to the Grassmanian/other classifying space.

There must also be plenty of examples outside algebraic topology, though this technique seems to be most popular there...

shenghao · Answer 3 · 2009-12-15 04:24:58Z UTC

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Devissage is a useful tool when proving something holds for a general class of objects, at least in algebraic geometry, like all schemes/stacks/morphisms.

answered Dec 15 at 4:24

shenghao

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What is devissage? – Ilya Grigoriev Dec 15 at 18:13

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I don't know if there is a formal definition. For me it roughly means that, when proving something in general, one reduces step by step to special cases. Some people say that it's an important feature in Grothendieck's style proofs. Here's an "example". Suppose we want to prove something for any morphism $f:X\to Y$ of schemes of finite type over a field k. One can check if this property is local in the sense that, if it's true for all fibers of $f,$ then it's true for $f.$ If it is local then we reduce to the case where Y is a point Spec k. (too long to fit in one comment; to be continued) – shenghao Dec 15 at 20:37

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Let U be any open subset of X with complement Z. Suppose that if the property holds for both U and Z, then it holds for X. Then we may shrink X to any open subset and use noetherian induction, so we can assume X is affine of equidimension d. There exists a map $X\to X'$ with fibers of dimension <=1, and dim X'=d-1, so by induction on d we reduce to two cases: X is a curve, or X is a point. Finally we prove these special cases. – shenghao Dec 15 at 20:39

Harrison Brown · Answer 4 · 2009-12-14 15:07:55Z UTC

Weil's "three columns": Number fields over $\mathbb{Q}$ behave like function fields of curves over finite fields which are related to the field of algebraic functions over $\mathbb{C}$. (This is far removed from my comfort zone, so please fix it if I'm off the mark.)

Orbicular · Answer 5 · 2009-12-14 15:15:49Z UTC

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If something does not hold, make it true! Examples: - Sobolev spaces (not necessarily differentiable functions satisfy differential equations) - distribution theory (think of identities involving the delta "function") - no converging? take the closure of your vector space (analysis) or compactify your space (geometry)

answered Dec 14 at 15:15

Orbicular

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Related to that, "Look in a bigger bag." For example, to find an integer solution, first find a rational or even complex solution then try to show it's an integer. Or find a solution in a Sobolev space then prove the solution is actually a classical solution. – John D. Cook Dec 14 at 21:15

Another famous example of this is the notion of stacks: quotients of schemes by group actions do not necessarily exist? We make them exist by sheer brute force! (And it's funny that the definition of a stack also mimics the definition of a distribution somehow -- in both situations, the idea is to forget the object itself and only remember how it acts on some class of "test objects".) – Dan Petersen Dec 15 at 8:19

Harrison Brown · Answer 6 · 2009-12-14 14:56:09Z UTC

If you want to show that a graph has few edges, prove that not too many vertices can have large degree.

(The complementary statement is the main trick in the solution to this MO question, by way of example. It's also used in the proof of the Stanley-Wilf conjecture.)

Nick Salter · Answer 7 · 2009-12-14 17:48:08Z UTC

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You must exchange the order of summation in order to prove any identity involving multiple sums.

answered Dec 14 at 17:48

Nick Salter

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Gjergji Zaimi · Answer 8 · 2009-12-14 19:46:43Z UTC

Fermat's infinite descent (often appearing as Vieta-jumping). When solving certain simple Diophantine equations in $N$ one starts with an unknown solution and from it tries to find a "smaller" solution. We know at some point we must reach either a stable solution or a negative one. Then we can rebuild all solutions recursively. Example: Pell equations.

Norman Lewis Perlmutter · Answer 9 · 2010-01-13 06:40:28Z UTC

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I forget who this is attributed to, but someone said something like "A technique is a trick used twice."

answered Jan 13 at 6:40

Norman Lewis Perlmutter

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Charles Siegel · Answer 10 · 2010-01-13 07:22:59Z UTC

Look at flabbier objects. This seems to be especially useful in complex algebraic geometry. Hard to prove something for varieties? See if there's a version that's true for schemes. Or maybe Kahler manifolds. Or worse: stacks. Vector bundles giving you trouble? Try coherent sheaves. Try quasi-coherent sheaves. In fact, try complexes of them. This is really just a special case of "Generalize the question as far as you can" but in this specific case, it's rather clarifying, here are some examples in algebraic geometry:

It's hard to say anything about fundamental groups of complex projective varieties that isn't also true about compact Kahler manifolds. Perhaps the proof should focus on using the Kahler structure, when you're working on these.
Want to parameterize subvarieties of a projective variety? Tough, it doesn't work. SubSCHEMES, however, gives the Hilbert Scheme.
Proving things about ideals is often easier to do with modules in general

What are some slogans that express mathematical tricks?

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