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How to prove anything
HOW TO PROVE ANYTHING (source unknown)6-15-89
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."
"Intuitively obvious to the most casual observer."
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.
Now, the lab session: A group of scientists of all kinds gathered one day
to prove that all odd numbers were prime numbers.
The mathematician was first: "One is a prime number, three is a prime
number, five is a prime number, seven is a prime number. Thus by induction,
all odd numbers are prime."
Then came the physicist: "One is a prime number, three is a prime number,
five is a prime number, seven is a prime number. Nine is NOT a prime number.
Eleven is a prime nummber, thirteen is a prime number. Well, nine falls within
experimental error, so all odd numbers are prime."
Finally the chemist: "One is a prime number, three is a prime number, five
is a prime number, seven is a prime number, nine is a prime number . . .
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