.. and the classics
How mathematicians do it...
Combinatorists do it as many ways as they can.
Combinatorists do it discretely.
(Logicians do it) or [not (logicians do it)].
Logicians do it by symbolic manipulation.
Algebraists do it in groups.
Algebraists do it in a ring.
Algebraists do it in a field.
Analysts do it continuously.
Real analysts do it almost everywhere.
Pure mathematicians do it rigorously.
Topologists do it openly.
Topologists do it on rubber sheets.
Dynamicists do it chaotically.
Mathematicians do it forever if they can do one and can do one more.
Cantor did it diagonally.
Fermat tried to do it in the margin, but couldn't fit it in.
Galois did it the night before.
Möbius always does it on the same side.
Markov does it in chains.
Newton did it standing on the shoulders of giants.
Turing did it but couldn't decide if he'd finished.
How to prove it..
proof by example:
The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.
proof by intimidation:
"Trivial."
proof by vigorous handwaving:
Works well in a classroom or seminar setting.
proof by cumbersome notation:
Best done with access to at least four alphabets and special symbols.
proof by exhaustion:
An issue or two of a journal devoted to your proof is useful.
proof by omission:
"The reader may easily supply the details"
"The other 253 cases are analogous"
"..."
proof by obfuscation:
A long plotless sequence of true and/or meaningless syntactically related statements.
proof by wishful citation:
The author cites the negation, converse, or generalization of a theorem from the literature to support his claims.
proof by funding:
How could three different government agencies be wrong?
proof by eminent authority:
"I saw Karp in the elevator and he said it was probably NP-complete."
proof by personal communication:
"Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication]."
proof by reduction to the wrong problem:
"To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem."
proof by reference to inaccessible literature:
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.
proof by importance:
A large body of useful consequences all follow from the proposition in question.
proof by accumulated evidence:
Long and diligent search has not revealed a counterexample.
proof by cosmology:
The negation of the proposition is unimaginable or meaningless. popular for proofs of the existence of God.
proof by mutual reference:
In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to follow from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.
proof by metaproof:
A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.
proof by picture:
A more convincing form of proof by example. Combines well with proof by omission.
proof by vehement assertion:
It is useful to have some kind of authority relation to the audience.
proof by ghost reference:
Nothing even remotely resembling the cited theorem appears in the reference given.
proof by forward reference:
Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.
proof by semantic shift:
Some of the standard but inconvenient definitions are changed for the statement of the result.
proof by appeal to intuition:
Cloud-shaped drawings frequently help here.
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