TITLE: What are some slogans that express mathematical tricks?
QUESTION [40 upvotes]: 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.

REPLY [5 votes]: "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.

REPLY [3 votes]: 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!