More Proof Techniques

Today in my Intermediate Analysis class, I presented 14 different proof techniques that we will be using throughout the semester. While the students were surprised to see that there were that many different types of proof techniques, they realized that by this point that had seen and used every last one of them. That was reassuring. However, I neglected to mention that are a great many more (invalid) proof techniques that their professor will likely make use of. Here a few for your consideration.

Proof by vigorous handwaving: Works well in a classroom or seminar setting.
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 funding: How could three different government agencies be wrong?
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 omission: “The reader may easily supply the details.” “The other 253 cases are analogous.

Proof by deferral: “We’ll prove this later in the course.
Proof by picture: A more convincing form of proof by example. Combines well with proof by omission.
Proof by intimidation: “Trivial.
Proof by seduction: “Convince yourself that this is true!
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 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 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 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.
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 semantic shift: Some of the standard but inconvenient definitions are changed for the statement of the result.
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 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 reduction to the wrong problem: “To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem.
Proof by metaproof: A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.


4 thoughts on “More Proof Techniques

  1. You left out Proof by strenuous assertion: The author repeatedly claims, with great forcefulness and conviction, that the theorem MUST be true! It MUST! It MUST!

    I’m partial to proof by cumbersome notation, but then again I’m an algebraic topologist with a category theory fetish.


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