PROOF TECHNIQUES

written by Armen H. Zemanian, published in The Physics Teacher, May 1994.

The usual techniques for proving things are often inadequate because they
are merely concerned with truth. For more practical objectives, there are
other powerful - but generally unacknowledged - methods. Here is an
(undoubtedly incomplete) list of them:

Proof of Blatant Assertion: Use words and phrases like
"clearly...,""obviously...,""it is easily shown that...," and "as any fool
can plainly see..."

Proof by Seduction: "If you will just agree to believe this, you might get
a better final grade."

Proof by Intimidation: "You better believe this if you want to pass the
course."

Proof by Interruption: Keep interrupting until your opponent gives up.

Proof by Misconception: An example of this is the Freshman's Conception of
the Limit Process: "2 equals 3 for large values of 2." Once introduced, any
conclusion is reachable.

Proof by Obfuscation: A long list of lemmas is helpful in this case - the
more, the better.

Proof by Confusion: This is a more refined form of proof by
obfuscation. The long list of lemmas should be arranged into circular
patterns of reasoning - and perhaps more baroque structures such as
figure-eights and fleurs-de-lis.

Proof by Exhaustion: This is a modification of an inductive proof. Instead
of going to the general case after proving the first one, prove the second
case, then the third, then the fourth, and so on - until a sufficiently
large n is achieved whereby the nth case is being propounded to a soundly
sleeping audience.

More proof methods: Proof by passion: The author gives the proof with a lot
of passion,
expressive eyes and vigorous movements...

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 literature to support his claims.

Proof by funding: How could three different government agencies be
wrong?

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 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
in 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 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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Even more Proof Techniques

Methods for getting people to believe you (as good as, if not better than,
proof). A collection of proof techniques that will prove invaluable to
both mathematicians and members of the general public.

PROOF TECHNIQUE #1 - 'Proof By Induction'

    1. Obtain a large power transformer.
    2. Find someone who does not believe your theorem.
    3. Get this person to hold the terminals on the HV side of the
         transformer.
    4. Apply 25000 volts AC to the LV side of the transformer.
    5. Repeat step (4) until they agree with the theorem.

PROOF TECHNIQUE #2 - 'Proof By Contradiction'

    1. State your theorem.
    2. Wait for someone to disagree.
    3. Contradict them.

PROOF TECHNIQUE #3 - Fire Proof

    1. Summon all your inferiors for a departmental meeting.
    2. Present your theorem.
    3. Fire those who disagree.

PROOF TECHNIQUE #4 - The Famous Water Proof

    1. State your theorem.
    2. Wait for someone to disagree.
    3. Drown them.

    NB. This is closely related to the 'bullet' proof, but is easier
    to make look like an accident.

PROOF TECHNIQUE #5 - Idiot Proof

    1. State your theorem.
    2. Write exhaustive documentation with glossy colour pictures
         and arrows about which bit goes where.
    3. Challenge anyone to not understand it.

PROOF TECHNIQUE #6 - Child Proof

    1. State your theorem.
    2. Encapsulate it in epoxy and shape it into an ellipsoid.
    3. Put it in a jar with all the other proofs (one with one of
         those Press-to-Open lids).
    4. Give it to a professor and challenge him to open it.

PROOF TECHNIQUE #7 - Rabbit Proof

    1. Generate theorems at an altogether startling rate, much
         faster than anybody is able to refute them. Use up every
         body else's paper. Run away at the slightest sign of danger.
    2. Leave any crap in small, easily identified piles, in
         prominent places where you no longer are, and it cannot in
         fact be proven that you ever were.

PROOF TECHNIQUE #8 - Fool Proof

    1. State your theorem.
    2. Invite colleagues to comment.
    3. If they don't agree, exclaim loudly, "You Fools!"