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If the proof of a theorem is not immediately apparent, it may be because
you are trying the wrong approach. Below are some effective methods of
proof that may aim you in the right direction.
- Proof by Obviousness: "The proof is so clear
that it need not be mentioned."
- Proof by General Agreement: "All in
Favor?..."
- Proof by Imagination: "Well, We'll pretend its
true."
- Proof by Convenience: "It would be very nice if
it were true, so ..."
- Proof by Necessity: "It had better be true or
the whole structure of mathematics would crumble to the
ground."
- Proof by Plausibility: "It sounds good so it
must be true."
- Proof by Intimidation: "Don't be stupid, of
course it's true."
- Proof by Lack of Sufficient Time: "Because of
the time constraint, I'll leave the proof to you."
- Proof by Postponement: "The proof for this is
so long and arduous, so it is given in the appendix."
- Proof by Accident: "Hey, what have we
here?"
- Proof by Insignificance: "Who really cares
anyway?"
- Proof by Mumbo-Jumbo: " For any epsilon> 0
there exists a corresponding delta > 0 s.t. f(x)-L < epsilon
whenever x-a < delta"
- Proof by Profanity: (example omitted)
- Proof by Definition: "We'll define it to be
true."
- Proof by Tautology: "It's true because it's
true."
- Proof by Plagiarism: "As we see on page 238
..."
- Proof by Lost Reference: "I know I saw this
somewhere ..."
- Proof by Calculus: "This proof requires
calculus, so we'll skip it."
- Proof by Terror: When intimidation fails ...
- Proof by Lack of Interest: "Does anyone really
want to see this?"
- Proof by Illegibility: " ¥ ª Ð Þ þæ"
- Proof by Logic: "If it is on the problem sheet,
then it must be true."
- Proof by Majority Rule: Only to be used if General
Agreement is impossible.
- Proof by Clever Variable Choice: "Let A be the
number such that this proof works."
- Proof by Tessellation: "This proof is just the
same as the last."
- Proof by Divine Word: "And the Lord said, 'Let
it be true,' and it came to pass."
- Proof by Stubbornness: "I don't care what you
say! It is true!"
- Proof by Simplification: "This proof reduces to
the statement, 1 + 1 = 2."
- Proof by Hasty Generalization: "Well, it works
for 17, so it works for all reals."
- Proof by Deception: "Now everyone turn their
backs ..."
- Proof by Supplication: "Oh please, let it be
true."
- Proof by Poor Analogy: "Well, it's just like
..."
- Proof by Avoidance: Limit of Proof by Postponement
as t approaches infinity.
- Proof by Design: "If it's not true in today's
math, invent a new system in which it is."
- Proof by Intuition: "I just have this gut
feeling ..."
- Proof by Authority: "Well, Bill Gates says it's
true, so it must be."
- Proof by Vigorous Assertion: "And I REALLY MEAN
THAT!"
- Proof by A.F.K.T. Theorem: "Any Fool Knows
That!"
- Proof by Divine Intervention: "Then a miracle
occurs ..."
- 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."
- 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 by 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, in a personal
conversation.]"
- 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 show 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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