Fifty ways not to prove a theorem
Invalid techniques of proof
The author gives only the case n = 2 and suggests that
it contains most of the ideas of the general proof.
"Trivial."
Works well in a classroom or seminar setting.
Best done with access to at least four alphabets
and special symbols.
An issue or two of a journal devoted to your proof is
useful.
"The reader may easily supply the details."
"The other 253 cases are analogous."
"..."
A long plotless sequence of true and/or meaningless
syntactically related statements.
The author cites the negation, converse, or generalization
of a theorem from literature to support his claims.
How could three different government agencies be wrong?
"I saw Karp in the elevator and he said it was probably
NP-complete."
"Eight-dimensional colored cycle stripping is NP-complete
[Karp, personal communication]."
"To see that infinite-dimensional colored cycle stripping
is decidable, we reduce it to the halting problem."
The author cites a simple corollary of a theorem to be
found in a privately circulated memoir of the Slovenian
Philological Society, 1883.
A large body of useful consequences all follow from the
proposition in question.
Long and diligent search has not revealed a counterexample.
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.
A method is given to construct the desired proof. The
correctness of the method is proved by any of these techniques.
A more convincing form of proof by example. Combines well
with proof by omission.
It is useful to have some kind of authority in relation to
the audience.
Nothing even remotely resembling the cited theorem appears
in the reference given.
Reference is usually to a forthcoming paper of the author,
which is often not as forthcoming as at first.
Some standard but inconvenient definitions are changed for
the statement of the result.
Cloud-shaped drawings frequently help here.
We were asked in an exercise to proof this theorem. Hence, it must be
true.
The majority of [proper authority] believes that...
"Surely, you are not implying that X is false."
Y implies not X, therefore not Y implies X.
The theorem is true for N = 0, and if the theorem is true for N+1,
then it is true for N.
The computer told us that X is true.
There are two kinds of animals: mammals and birds. Mammals are
homeotherm. Birds are homeotherm. Therefore, all animals are
homeotherm.
The professor simply and conveniently writes down a variation on
quadratic reciprocity, despite the fact that the student pointed out
that 87 is actually not a prime modulus.
We don't have time to prove this...
The reader is left to do the proof as an exercise. Most commonly
found in textbooks.
Any experiment that could collect confounding data would violate
accepted ethical guidelines.
Schedule a talk to present results refuting our claims and we get an
injunction under the Digital Millennium Copyright Act or local
equivalent.
Typically factoring integers or finding logarithms in finite groups.
This method of proof is one of the two pillars of modern cryptography.
We offered a $10,000 prize for anyone who could solve this puzzle and
no one has come forward with a correct answer. The other pillar.
If you point out an error in my lengthy and incoherent proof, I will
send you an even longer, more impenetrable manuscript that purports
to correct the mistake.
"The paper contains an outline of the proof; an extended version of the
paper will contain the full proof once it is finished."
"In the interest of time, I will skip the details..."
"I will spare you the details and move on to the main result."
"... will present the theorem's proof after recess..."
-between 10 and 15 minutes pass-
"...as proven before recess, we can now ..."
If it sounds right, it is right.
In which a special form of an equation, definition, or technique used
by one science is applied in a completely unrelated field. Best used
by engineering undergraduates in pure mathematical courses, where the
assignments will be graded by computer science undergraduates.
In which a phrase of the form "the x of y" becomes "the y of x" three
paragraphs later.
"Your question is beyond the scope of this course. However, next
semester, if you want to explore this more thoroughly..."
There are two variants that I can think of. (a) As you have it; that
is, you use different definitions from the normal ones (preferably,
not obviously inequivalent) and prove a theorem that appears to be,
but isn't, interesting. (b) Change the meanings of the terms in the
course of the proof. This variant is best used in combination with
the following method of proof.
The definitions are not so well-defined
that the reader can tell exactly what they mean.
There are so many errors that they cancel
each other out. (Students often do this.)
There are so many errors that the reader
can't tell whether the conclusion is proved or not, so is forced to
accept the claims of the writer.