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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:

  1. 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.
  2. 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.
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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 '09 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 '09 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 '09 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 '10 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 '10 at 11:01

21 Answers 21

Jacobi's famous quote that "one must always invert." He had elliptic integrals in mind.

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  1. Pick a random example.

  2. If you add lots of small and reasonably independent things together then the result will be highly concentrated about its mean.

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"It is easy to prove existence when there is only one, or when there are many"

explanation:

If there is only one object with a certain property, you can sometimes use it to define it. For example, in geometric situations, you can sometimes define it locally and glue the patches since uniqueness guarantees compatibility on overlaps. It suggests that you should try proving uniqueness before proving existence and if uniqueness fails, maybe you should add constraints (thus, paradoxically, adding constrains can help in proving existence). On the other hand, sometimes it is easier to prove that there are many than to point out one specific example (transcendental numbers, continues nowhere differentiable functions,...). Therefore, you may want to seek for the right notion of "many" in your universe (cardinality, measure, "topological bigness" like the baire property,...) and try to prove that actually there are "few" objects that don't have the required property.

comment: This relates to the answer saying that when you can't avoid making a choice, make all of them simultaneously. This happens when there are more than one, but not many...

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"If you count something two different ways, you get the same result." This is related to the trick of changing the order of integration (or summation) discussed above, but discrete and more general.

This method is used all the time in combinatorics. I think it has also been phrased differently, but I don't remember the exact phrasing.

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"When in doubt, differentiate." I've heard this attributed to Chern.

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One of the slogans in T. W. Körner's book Fourier Analysis that is definitely in the harmonic analyst's toolbox: The function $f*g$ has the good properties both of f and g.

An example of its use is in approximating functions by trigonometric polynomials: convolving the function with any trigonometric polynomial gives you a trigonometric polynomial, and if you pick the polynomial carefully the resulting function will have similar properties to the original one.

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With one exception: support in physical space. If $f$ has compact support and $g$ does not, the convolution does not have compact support. Of course, another way of looking at it is that having compact support is such an unstable property that it is actually not good in harmonic analysis. –  Willie Wong Oct 1 '10 at 15:06

If you have to chose some auxiliary object and that object is not unique, it's better to make all choices simultaneously.

I think there are many examples of this, but for me it first hit home when I learned about crystalline cohomology. There you want to lift varieties in positive characteistic to characteristic zero. Locally there are many nonisomorphic lifts, and rather than picking one, it's better to work with the category of all of them. I've absorbed this lesson pretty fully, to the point where I don't need to remind myself of it, but at first it seemed revolutionary.

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I'm not sure if this is a bit too general, but it is a slogan/heuristic that I find very useful and that I think most people will be able to come up with plenty of examples of:

"Extremalities always arise from symmetry."

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I would personally change 'always' to 'often'. Life for geometric analysts will be somewhat easier if this were actually a theorem. In the theoreical physics literature this is related to the so-called Coleman's Principle. However, it was shown by Kapitanski and Ladyzhenskaya that unless one makes additional assumptions, this principle is generally incorrect. ams.org/mathscinet-getitem?mr=711846 –  Willie Wong Oct 1 '10 at 15:14

"Think homologically, prove cohomologically!" definitely sounds like a slogan. One argument for this is that homology has a nice explanation in terms of geometry, think singular simplices or cells, so you can think about a space in terms of its cellular homology. When proving things you might want to have more structure around, like a product, and this is where cohomology comes in.

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The best way to solve a problem is to define it out of existence.

Typical example: Weil constructed abelian varieties over finite fields, and at first he did not know if these were varieties because it was not clear that they were projective. Weil defined this problem out of existence by changing the definition of variety and inventing abstract varieties.

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There are two interesting tricks in K-theory / operator algebras / homotopy theory - one attached to an amusing slogan and the other with an amusing name - that I think foot the bill.

The first is "uniqueness is a relative form of existence", due apparently to Shmuel Weinberger. This slogan seems to appear frequently in operator theory. Take, for example the problem of proving that K-theory commutes with direct limits (say, of C* algebras $A_1 \subseteq A_2 \subseteq \ldots \subseteq A$). There are two components to the proof: surjectivity (the "existence" part) which amounts to showing that every element of $K_0(A)$ lies in the image of some $K_0(A_j) \to K_0(A)$, and injectivity (the "uniqueness" part) which involves proving that if two elements of $K_0(A_j)$ are equivalent in $K_0(A)$ then they are equivalent in $K_0(A_j)$. Once you have proven existence you can verify uniqueness by joining representatives of your chosen $K_0(A_j)$ classes by a homotopy in the space of generators for $K_0(A)$ and then use your existence argument to lift to a homotopy in $A_j$. In other words, prove uniqueness by applying your existence argument to a pair.

The second is the (in)famous "Eilenberg Swindle" which seems to come up everywhere. I first encountered it in K-theory, but I think the canonical example is the argument which proves that the $n$-sphere is prime with respect to connected sum (which I will denote +). Suppose that $M$ and $N$ are manifolds such that $M + N = S^n$. We have that $(M + N) + (M + N) + (M + N) + \ldots$ is homeomorphic to $\mathbb{R}^n$ (it is a cylinder with the left opening glued shut), and similarly so is $(N + M) + (N + M) + \ldots$. Since $M + (N + M) + \ldots = (M + N) + (M + N) + \ldots$, we have shown that $M + \mathbb{R}^n = \mathbb{R}^n$ which forces $M$ to be homeomorphic to $S^n$.

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This is definitely older than Shmuel, but i guess you are referring specifically to the quote. The main point is that uniqueness in homotopy theory is just saying everything you want to construct is connected by a homotopy. So you need that homotopy to exist, but you have an existence result that you can usually apply to construct your homotopy. Also think of the example provided by whitney's embedding theorem and how it shows any two embeddings are homotopic. –  Sean Tilson Sep 26 '10 at 6:02

A perfect example are the twelve heuristics listed on page 1 of L. Larson, "Problem solving through problems": http://books.google.com/books?id=qFNZIUQ_MYUC&lpg=PP1&dq=larson%20problem%20solving&pg=PA1#v=onepage&q&f=false

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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:

  1. 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.
  2. Want to parameterize subvarieties of a projective variety? Tough, it doesn't work. SubSCHEMES, however, gives the Hilbert Scheme.
  3. Proving things about ideals is often easier to do with modules in general
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can you elaborate on 3? –  Ho Chung Siu Jan 13 '10 at 17:04
    
Well, that's pretty much one of the main themes of commutative algebra books: that looking at ideals is the wrong point of view, really they're just submodules of the free module on one generator, and many (most, perhaps) results about ideals are actually true about modules, and how to prove them is easier to see when you realize just what the right way to look at things is. –  Charles Siegel Jan 13 '10 at 17:10

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

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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.

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What is devissage? –  Ilya Grigoriev Dec 15 '09 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 '09 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 '09 at 20:39

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...

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Along these lines, replace a subset of a finite set by an element of a vector space over GF(2). –  Harrison Brown Dec 15 '09 at 2:50

The analyst's toolbox consists of three things:

  1. The Cauchy-Schwarz inequality
  2. Changing the order of integration/summation
  3. Integration by parts

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

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You forgot 4. Adding and substracting something. –  Mariano Suárez-Alvarez Jan 13 '10 at 7:37
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Would you allow me the principle that if the average of some numbers is at least C then one of the numbers must be at least C? –  gowers Sep 23 '10 at 21:42
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An analyst once told me: "When in doubt, integrate by parts." –  Micah Milinovich Sep 23 '10 at 22:42
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I've also heard the following cited as a tool: if $a \leq b + \epsilon$ for all $\epsilon > 0$, then $a \leq b$. –  S. Carnahan Sep 24 '10 at 3:02
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When Peter Lax went to receive the national medal of science, he was asked by the other recipients about his merits. His answer was (apocryph) I integrated by parts. –  Denis Serre Apr 7 '11 at 9:33

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

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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)

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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 '09 at 21:15
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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 '09 at 8:19

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.)

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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.)

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is there away of enlarging this to other fields? or rather, how should it be phrased? –  Sean Tilson Sep 26 '10 at 6:04
    
Something like "If the average is small, and you have controlled the large outliers, then everything must be small" but that's not very snappy. –  Matthew Daws Oct 1 '10 at 14:04

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