School of Mathematics and Statistics
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Faculty of
Engineering and Mathematical Sciences |
University of Western Australia
The following is a list of some common proof techniques that are
often extremely useful. I received the original list long time ago
via e-mail. Recently, or rather over time,
Michael Ashley,
Deepak Azad,
Roman Beslik,
Atanas Boev,
John Chase,
Maxwell Davenport,
Christophe de Dinechin,
Hedi Driss,
Dan Echlin,
Ron Garret,
Manuel Hoffmann,
Arthur Keller,
Vivek Joy Kozhikkottu,
John McGoff,
Scott A. McIntyre,
John Norberg,
Filip Opálený,
Alex Papanicolao,
József Pék,
Arnold G. Reinhold,
Noemi Reynolds,
Greg Rose,
Lucas Scharenbroich,
John Snider,
Ki Song,
Moritz Voss,
David N. Werner and
Thomas Zaslavsky suggested some additions. I also added my own experiences
from teaching techniques of proofs in a first year mathematics units
for several years.
Unfortunately all these proof techniques are invalid and, hence,
one should not use them in assignments, workshops, exams,
papers, etc.
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.
The negation of the proposition is unimaginable or meaningless.
Popular for proofs of the existence of God.
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.
This works best with chalk on a blackboard. E.g., if at some stage in
the proof you have "(x^4)/x", erase the first parenthesis and "4)/x"
and replace the latter with "3", yielding "x^3". After no more than a
few moments of writing and erasing, the final result will appear on
the blackboard, with no extraneous working.
[I learnt this from an applied maths lecturer at ANU in the late
1970's; and he actually used it! He would write using chalk with one
hand, and an eraser in the other.]
"I bet a $1000 that this is true, are you willing to take this bet?" Proof
is complete if no one takes the bet.
"The proof easily can be found in the literature." "The literature" is
so vast that someone easily can spend a year looking for just one proof.
The equation holds true for values of much larger than x.
Because the proof looks good and is typed in LaTeX, it must be right.
Often employed in classes of few students when they decide the question is
too hard.
Where a student does a lot of handwaving, but uses their in-depth
knowledge of things closely related to the proof to convince the
marker they know what they are talking about. The details are left as
an exercise for the marker who knew 'what they meant'.
The majority of [proper authority] believes that...
The cost would be prohibitive if X was proven false...
God told us that X is true (frequent among creationists)
“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.
Needs a loud assertive voice, muscled jaws can increase its efficiency.
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...
This is a refinement of "proof by anecdote" where the data point cited
has a strong negative emotional impact. This is a popular technique
in politics and product liability lawsuits.
Most prominently used to help acquit O.J. Simpson of murder. "If it
doesn't fit you must acquit."
I often get from first semester students:
They draw a little figure of how they think, e.g. a graph looks like,
and deduce facts by this illustration. This way often special cases
(like empty graph, isolated nodes, ...) are ignored.
Best used when giving the proof verbally. cf
http://en.wikipedia.org/wiki/Wikipedia:How_many_legs_does_a_horse_have%3F
and
A cow have four more legs than no cow. No cow has five legs.
Therefore, a cow has nine legs.
Show it's true for n=1 and for n+1 assuming n. But the induction
step works only for n>1. cf. Proof that all horses are the same
color.
Slip by the reader that you have assumed both P and not P. Then you
can deduce any Q.
X is true because of Y, but you haven't proven Y yet and the proof is
so long it's hard to notice.
"Because I said so."
Have many people claim it's true to the media, and the media will
start believing it.
Used to deny the hazards of cigarette smoke for many years, and now
climate change deniers. Made easier with media being complicit by
always needing to present contrary points of view to nearly any issue.
Don't trust it because the one who gave the proof will benefit
financially from its acceptance.
Used often in cases of free energy claims or most quantum mechanical
possibilities. Closely related to proof by common sense and proof by
popularity
This applies when the author cites lack of pages in a journal or
students in an exam.
Used by a student who really needs to pass a course.
Work forward from what you know on one page. Then work backwards from
the answer on the other side. Nobody notices.
An entire body of (sort-of) elected officials is more correct than all
of the known laws of physics, math and science as a whole.
I thought I would share a comment regarding something one of my
professors would say a lot. He never used it as a proof, but rather an
introduction of a fact:
"If it walks like an elephant, talks like an elephant, it must be an elephant."
My Linear Models practical lesson tutor's very popular proof
technique:
"...and the proof why this is true we'll show after the break."
"As we proved before the break, this theorem is true for..."
The reader is left to do the proof as an exercise. Most commonly
found in textbooks.
Theorem 12.3's proof is beyond the scope of your course, however, it
is possible to prove it using more advanced Mathematics.
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.
Using an appeal based on one's ‘nice’
personality. “You HAVE to believe me because I am awfully
sweet. I don't say things that are not true — that would not be
nice.”
Claiming that if a statement is not true then terrible things would
happen to the world and/or humanity. “If it isn't true then
humanity is so much the poorer for it. People are better off believing
it.”
Misrepresenting an opposing argument and then pointing out what is
wrong with that argument based on that representation. For
example,
Person A: “The vast majority of primes are odd.”
Person B: “Person A says all primes are odd. But 2 is prime and
not odd so A's statement is wrong.”
Stating an assumption and using deductive reasoning based on that
assumption to prove that it is true. Basically a ‘If it is true
then it is true’ type of argument. A gross distortion of proof
by contradiction. For example: “Assume all primes are odd
numbers. If all primes are odd numbers then no prime can be
even. Therefore there are no even primes.”
“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.”
The premise of this proof technique is essentially “Nobody has ever
seen one, so they don't exist.” Alternatively, “All of the items I
tried worked, so this must be right.”
I was surfing the web for a list of invalid proofs and I ran into your
website. It's pretty neat, and it reminded me of a student who tried
to solve the following problem: “Find all isomorphism classes of
groups of order 28.”
The student basically tried to enumerate every single group of that
order, and he/she basically went through each group to show that the
property holds for them.
The grader wrote on the page: “Proof by Exhaustion of Grader.”
“... 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.
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Author: Berwin A Turlach
Date Last modified:Tue Jan 17 14:28:25 AWST 2017
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URL: http://www.maths.uwa.edu.au/~berwin/humour/invalid.proofs.html