|
HOW
TO PROVE IT
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 the literature to support his claims. proof by funding:
How could three different government agencies be 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,
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 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. Dana Angluin, Sigact News ,
Winter-Spring 1983, Volume 15 #1
A quiet little
man was brought before a judge. The judge looked down at the man and
then at the charges and then down at the little man in amazement. "Can
you tell me in your own words what happened?" he asked the man.
"I'm a
mathematical logician dealing in the nature of proof."
"Yes, go on,"
said the astounded judge.
"Well, I was at
the library and I found the books I wanted and went to take them out.
They told me my library card had expired and I had to get a new one. So
I went to the registration office and got in another line. And filled
out my forms for another card. And got back in line for my card."
"And?" said the
judge.
"And he asked
'Can you prove you are from New York City?' ...So I stabbed him."
There was a
logician who saw a sign on his way to fish that read, "All the worms
you want for $1.00." He stopped his car and ordered $2.00 worth.
The book Dynamic
Programming by Richard Bellman is an important, pioneering
work in which a group of problems is collected together at the end of
some chapters under the heading "Exercises and Research Problems," with
extremely trivial questions appearing in the midst of deep, unsolved
problems. It is rumored that someone once asked Dr. Bellman how to tell
the exercises apart from the research problems, and he replied: "If you
can solve it, it is an exercise; otherwise it's a research problem."
 A mathematician
is a machine for turning coffee into theorems. 
A conjecture
both deep and profound
Is whether a circle is round.
In a paper of Erdös
Written in Kurdish
A counterexample is found.
Approximately
ten excuses for not doing homework:
I accidentally divided by zero and my paper burst into flames.
I could only get arbitrarily close to my textbook. I couldn't actually
reach it.
I have the proof, but there isn't room to write it in this margin.
I was watching the World Series and got tied up trying to prove that it
converged.
I have a solar powered calculator and it was cloudy.
I locked the paper in my trunk but a four-dimensional dog got in and
ate it.
I couldn't figure out whether I am the square of negative one or I am
the square root of negative one.
I took time out to snack on a doughnut and a cup of coffee, and then I
spent the rest of the night trying to figure which one to dunk.
I could have sworn I put the homework inside a Klein bottle, but this
morning I couldn't find it.
After
Receiving an Invitation to a Mathematicians' Ball:
 Augustin Louis
Cauchy said he surely will managed to integrate well with everyone.
 David
Hilbert was afraid he will be pretty spaced out for most of the party.
 Paul
Erdös asked: "Are epsilons invited too?"
 John
Forbes Nash insisted on playing n-person zero sum games.
 Zeno of
Elea said he will come with two friends - Achilles and the tortoise.
 Bertrand
Russell was wondering: "If the cook only cooks for the guests, who
cooks for the cook?"
 Kurt
Gödel insisted that the invitation is incomplete and never
will be.
Pick-Up
Lines to use on Mathematics Chicks
 You fascinate me
more than the Fundamental Theorem of Calculus.
 Are you a
differentiable function? Because I'd like to be tangent to your curves!
 You and I
would add up better than a Riemann sum.
 My love for
you is a monotonic increasing function of time.
 Wanna come
back to my room and see my copy of Euclid's "Elements"?
 I am
equivalent to the Empty Set when you are not with me.
The
Dictionary : what mathematics professors say and what
they mean by it
Clearly: I don't
want to write down all the "in-between" steps.
Trivial: If I have to show you how to do this, you're in the wrong
class.
It can easily be shown: No more than four hours are needed to prove it.
Check for yourself: This is the boring part of the proof, so you can do
it on your own time.
Hint: The hardest of several possible ways to do a proof.
Brute force: Four special cases, three counting arguments and two long
inductions.
Elegant proof: Requires no previous knowledge of the subject matter and
is less than ten lines long.
Similarly: At least one line of the proof of this case is the same as
before.
Two line proof: I'll leave out everything but the conclusion, you can't
question 'em if you can't see 'em.
Briefly: I'm running out of time, so I'll just write and talk faster.
Proceed formally: Manipulate symbols by the rules without any hint of
their true meaning.
Proof omitted: Trust me, It's true.
Mathematics
Revisited
 Life is
complex. It has real and imaginary components.
 What keeps
a square from moving? Square roots, of course.
 The law of
the excluded middle either rules or does not rule.
 In the
topological hell the beer is packed in Klein's bottles.
 To a
mathematician, real life is a special case.
 I heard
that parallel lines actually do meet, but they are very discrete.
 In modern
mathematics, algebra has become so important that numbers will soon
only have symbolic meaning.
 Some say
the pope is the greatest cardinal.
But others insist this cannot be so, as every pope has a successor.
How
mathematicians do it..
Combinatorists
do it as many ways as they can.
Combinatorists do it discretely.
(Logicians do it) or [not (logicians do it)].
Logicians do it by symbolic manipulation.
Algebraists do it in groups.
Algebraists do it in a ring.
Algebraists do it in a field.
Analysts do it continuously.
Real analysts do it almost everywhere.
Pure mathematicians do it rigorously.
Topologists do it openly.
Topologists do it on rubber sheets.
Dynamicists do it chaotically.
Mathematicians do it forever if they can do one and can do one more.
Cantor did it diagonally.
Fermat tried to do it in the margin, but couldn't fit it in.
Galois did it the night before.
Möbius always does it on the same side.
Markov does it in chains.
Newton did it standing on the shoulders of giants.
Turing did it but couldn't decide if he'd finished.
You Might Be a Mathematician if...
you are fascinated by the equation  .
you know by heart the first fifty digits of  .
you have tried to prove Fermat's Last Theorem.
you know ten ways to prove Pythagoras' Theorem.
your telephone number is the sum of two prime numbers.
you have calculated that the World Series actually diverges.
you are sure that differential equations are a very useful tool.
you comment to your wife that her straight hair is nice and parallel.
when you say to a car dealer "I'll take the red car or the blue one"
you must add "but not both of them."
|
How many
mathematicians does it take to change a light bulb?
None. It's left to the reader as an exercise.
 None. The
answer is intuitively obvious.
 One. He gives
it to four programmers, thereby simplifying the problem to a previous
question .
How many
numerical analysts does it take to change a light bulb?
 3.9967 (after
six iterations).
How many
mathematical logicians does it take to change a light bulb?
 None. They
can't do it, but they can easily prove that it can be done.
How many
classical geometers does it take to change a light bulb?
 None. You
can't do it with a straight edge and a compass.
How many
analysts does it take to change a light bulb?
 Three. One to
prove existence, one to prove uniqueness and one to derive a
nonconstructive algorithm to do it.
How many number
theorists does it take to change a light bulb?
 I don't know
the exact number, but I am sure it must be some rather elegant prime.
Back
to Index page.
|
