What are some slogans that express mathematical tricks?

Question

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.

Answer 1

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)

Answer 2

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.

Answer 3

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.)

Answer 4

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

Answer 5

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...

Answer 6

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.

Answer 7

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.)

Answer 8

"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.

Answer 9

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

Answer 10

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.

Answer 11

"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.

Answer 12

"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...

Answer 13

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 $S^n$.

Answer 14

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

Answer 15

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

Answer 16

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.)

Answer 17

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.

Answer 18

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.

Answer 19

"It is better to study a nice category of ugly objects than an ugly category of nice objects"

Examples:

• Say you want to study projective varieties, perhaps smooth, you better study them into the category of schemes which is nicer (you are allowed to perform more constructions) but some objects look weird at first glance.

• You want to study (finite dimensional) vector bundles over a variety but is is convenient to consider the category of coherent sheaves that it is abelian. Moreover, if you want to take infinite limits, then you generalizer further to the category of quasi-coherent sheaves.

• The previous example can be read algebraically: the category of projective finitely generated modules is not abelian but finite presentation modules is.

Answer 20

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."

Answer 21

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

1. Search for a pattern.
2. Draw a figure.
3. Formulate an equivalent problem.
4. Modify the problem.
5. Choose effective notation.
6. Exploit symmetry.
7. Divide into cases.
8. Work backward.
9. Argue by contradiction.
10. Pursue parity.
11. Consider extreme cases.
12. Generalize.

Answer 22

One of my favourite such slogans is "Just do it". This is very well covered on the Tricki:

http://www.tricki.org/article/Just-do-it_proofs

but as you say it is not as active as MO. I first discovered this trick when trying to solve a problem which is now a favourite of mine: Does there exist an enumeration $q_1,q_2,\dots$ of $\mathbb{Q}$ such that the series $$ \sum_{n=1}^{\infty} (q_{n+1} - q_n)^2 $$ converges?

Writing the problem here already gives away that the answer is 'yes' I won't write out a solution but there's a certain kind of solution which, once you've thought of it is very much: Well...just...do it. Just enumerate them so it converges!

Answer 23

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

Answer 24

"Yes and No are the smallest possible answers yet they need the most thinking to be done " !!

Answer 25

Not really a slogan, but here is something I came up with about 4 years ago at the end of a talk on combinatorial nullstellensatz when someone from the audience asked about the case when equality is achieved in Cauchy-Davenport inequality.

Whenever you see a sum-set's size being bounded, a good guess about the case when equality is achieved is always arithmetic progressions.

I almost even forgot about it, but just recently I was told by someone who watched the lecture that in general they found the above upshot useful, so I thought I might include it here as well.

Answer 26

I don't know if it fits here, but saying : "Nothing exists if it means something not that of a minimum or a maximum" ........ (I think I read this was by Euler)