Here's to pure mathematics, may it be of no use to anyone!
-- Attributed to G. H. Hardy

A mathematician is a machine for converting coffee into theorems.
-- Paul Erdos

A mathematician confided
That a Moebius strip is one-sided.
You' get quite a laugh
If you cut it in half,
For it stays in one piece when divided.

A mathematician named Klein
Thought the Moebius Band was divine.
Said he, "If you glue
The edges of two
You get a weird bottle like mine.

A challenge for many long ages
Had baffled the savants and sages.
Yet at last came the light:
Seems Fermat was right--
To the margin add 200 pages.

-- Paul Chernoff

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 pointless 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 Kluge in the elevator and he said it was probably NP- complete.'

proof by personal communication:
'Eight-dimensional colored cycle stripping is NP-complete
[Kluge, 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.

HOW TO PUT AN ELEPHANT INTO A REFRIGERATOR:

Analysis:
1) Differentiate it and put into the refrig. Then integrate it in the refrig.
2) Redefine the measure on the refrigerator (or the elephant).
3) Apply the Banach-Tarsky theorem.

Number theory:
1) First factorize, second multiply.
2) Use induction. You can always squeeze a bit more in.

Algebra:
1) Step 1. Show that the parts of it can be put into the refrig.
Step 2. Show that the refrig. is closed under the addition.
2) Take the appropriate universal refrigerator and get a surjection from refrigerator to elephant.

Complex analysis:
Put the refrig. at the origin and the elephant outside the unit circle.
Then get the image under the inversion.

Numerical analysis:
1) Put just its trunk and refer the rest to the error term.
2) Work it out using the Pentium.

Linear algebra:
1) Put just its basis and span it in the refrig.
2) Show that 1%

Statistics:
1) Bright statistician:
Put its tail as a sample and say "Done."

2) Dull statistician:
Repeat the experiment pushing the elephant to the refrig.

3) Our NEW study shows that you CAN'T put the elephant in the refrigerator.

Topology:
1) Have it swallow the refrigerator and then turn itself inside out.
2) Use a Klein bottle refrigerator.
3) The elephant is homeomorphic to a smaller elephant.
4) The elephant is compact, so it can be put into a finite collection of refrigerators. That's usually good enough.
5) The property of being inside the refrigerator is hereditary. So, take the elephant's mother, cremate it, and show that the ashes fit inside the refrigerator.
6) For those who object to method 3 because it's cruel to animals. Put the elephant's baby in the refrigerator.

Algebraic topology:
Replace the interior of the refrigerator by its universal cover, R3.

Inflation Explained! Mathematics, despite the lack of a Nobel prize in the field, has done what economics couldn't.
Proof:
1\$ = 100¢
= (10¢)2
= (0.1\$)2
= 0.01\$
= 1¢

This one is scary in that I have seen Ph D's in math who were unable to see what was wrong with this one. Actually I am crossposting this to sci.physics because I think that the latter makes a very nice introduction to the importance of keeping track of your dimensions...
-- Benjamin. J. Tilly

(Those physicists I've shown it to figured it out within a minute. Ed)

Theorem: 1 = -1
Proof by cross multiplication:
1 / -1 = -1 / 1

sqrt[1 / -1] = sqrt[-1 / 1]

sqrt[1]   sqrt[-1]
------- = -------
sqrt[-1]   sqrt[1]

-1 = 1

Theorem : 3 = 4
Proof:
Suppose that a + b = c
This can also be written as: 4a - 3a + 4b - 3b = 4c - 3c
After reorganizing: 4a + 4b - 4c = 3a + 3b - 3c
Take the constants out of the brackets: 4*(a+b-c) = 3*(a+b-c)
Remove the same term left and right: 4 = 3

Theorem: 4 = 5
Proof:
16 - 36 = 25 - 45
42 - 9*4 = 52 - 9*5
42 - 9*4 + 81/4 = 52 - 9*5 + 81/4
(4 - 9/2)2 = (5 - 9/2)2
4 - 9/2 = 5 - 9/2
4 = 5
-- Kevin D. Quitt

Theorem: a cat has nine tails.
Proof:
No cat has eight tails.
A cat has one tail more than no cat.
Therefore, a cat has nine tails.

Theorem: All numbers are equal.
Proof:
Choose arbitrary a and b, and let t = a + b.
a + b = t
(a + b)(a - b) = t(a - b)
a2 - b2 = ta - tb
a2 - ta = b2 - tb
a2 - ta + (t2)/4 = b2 - tb + (t2)/4
(a - t/2)2 = (b - t/2)2
a - t/2 = b - t/2
a = b
So all numbers are the same, and math is pointless.

Theorem : All numbers are equal to zero.
Proof: Suppose that a = b. Then
a = b
a2 = ab
a2 - b2 = ab - b2
(a + b)(a - b) = b(a - b)
a + b = b
a = 0

Theorem:
All positive integers are interesting.
Proof:
Assume the contrary. Then there is a lowest non-interesting positive integer. But, hey, that's pretty interesting! A contradiction!
QED (Quod Erat Demonstrandum: [Latin, rough translation: Who Cares?])

Theorem: e=1
Proof:
2*e = f
2(2pii)ie(2*pi*i) = f(2*pi*i)
e(2pii) = 1
so:
2(2pii) = f(2*pi*i)
2=f
thus:
e=1

Theorem: ln(2) = 0
Proof:
Consider the series equivalent of ln 2:
ln 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ...
Rearrange the terms:
ln 2 = (1 + 1/3 + 1/5 + 1/7 ...) - (1/2 + 1/4 + 1/6 + 1/8 ...)
Thus:
ln 2 = (1 + 1/3 + 1/5 + 1/7 ...) + (1/2 + 1/4 + 1/6 + 1/8 ...) - 2 * (1/2 + 1/4 + 1/6 + 1/8 ...)
Combine the first two series:
ln 2 = (1 + 1/2 + 1/3 + 1/4 + 1/5 ...) - (1 + 1/2 + 1/3 + 1/4 + 1/5 ...)
Therefore:
ln 2 = 0

Theorem: log(-1) = 0
Proof:
a) log[(-1)2] = 2 * log(-1)
On the other hand:
b) log[(-1)2] = log(1) = 0
Combining a) and b) gives:
2 * log(-1) = 0
Divide both sides by 2:
log(-1) = 0

Theorem: n = n+1
Proof:
(n+1)2 = n2 + 2*n + 1
Bring 2n+1 to the left: (n+1)2 - (2n+1) = n2
Subtract n(2n+1) from both sides and factoring, we have:
(n+1)2 - (n+1)(2n+1) = n2 - n(2n+1)
Adding 1/4(2n+1)2 to both sides yields:
(n+1)2 - (n+1)(2n+1) + 1/4(2n+1)2 = n2 - n(2n+1) + 1/4(2n+1)2
This may be written: [(n+1) - 1/2(2n+1) ]2 = [ n - 1/2(2n+1) ]2
Taking the square roots of both sides: (n+1) - 1/2(2n+1) = n - 1/2(2n+1)
Add 1/2(2n+1) to both sides: n+1 = n

Theorem: 1 = 2
Proof:
Let a = b.
Then
a2 = ab
a2 + a2 = a2 + ab
2a2 = a2 + ab
2a2 - 2ab = a2 + ab - 2ab
2a2 - 2ab = a2 - ab.
This can be rewritten as
2(a2 - ab) = 1(a2 - ab).
Dividing both sides by a2 - ab gives
2 = 1.

A proof that girls are evil.
Given: Girls require time and money.
Girls = Time * Money
Given: Time is money.
Time = Money
Therefore:
Girls = Money * Money = (Money)2
It is well known that "Money is the root of all evil", so
Money = sqrt(Evil)
It then follows that
Girls = (sqrt(Evil))2
So:
Girls = Evil.
QED.

(1) Alexander the Great was a great general.
(2) Great generals are forewarned.
(3) Forewarned is forearmed.
(4) Four is an even number.
(5) Four is certainly an odd number of arms for a man to have.
(6) The only number that is both even and odd is infinity.
Therefore, Alexander the Great had an infinite number of arms.

Every Horse has an Infinite Number of Legs (proof by intimidation):
Horses have an even number of legs.
Behind they have two legs, and in front they have fore-legs.
This makes six legs, which is certainly an odd number of legs for a horse.
But the only number that is both even and odd is infinity.
Therefore, horses have an infinite number of legs.
Now to show this for the general case, suppose that somewhere, there is a horse that has a finite number of legs.
But that is a horse of another color, and by the [above] lemma ["All horses are the same color"], that does not exist.
Q. E. D.
(Abbreviation for Quod Erat Demonstrandum, Latin for "Who cares?")

For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. The ends of things and their beginnings are impregnably concealed from him in an impenetrable secret. He is equally incapable of seeing the nothingness out of which he was drawn and the infinite in which he is engulfed.
-- Blaise Pascal

Proofs that odd numbers are prime:
Mathematician:
1 is prime, 3 is prime, 5 is prime, 7 is prime,
therefore, by induction, all odd numbers are prime.
Physicist:
1 is prime, 3 is prime, 5 is prime, 7 is prime,
9 is a bad data point, 11 is prime, 13 is prime...
Engineer:
1 is prime, 3 is prime, 5 is prime, 7 is prime,
9 is approximately prime, 11 is prime, 13 is prime...
Computer Scientist:
1 is prime, 1 is prime, 1 is prime, 1 is prime, 1 is prime, ...

Proof that Jesus was an Irishman:
1) He lived at home until he was 30.
2) The night before he died, he went out drinking with his friends.
3) His mother thought he was God.
4) He thought his mother was a virgin.
(Catholics: It's not a sin to laugh at this, and only venal to enjoy it.)

God is real. All Others are integers. Add God to Other;
The result is real but not necessarily rational.

-- A proof that GOD + RELIGION is possibly crazy

Newton's proof of the law of [planets moving in elliptical orbits] marked the culmination of the Scientific Revolution.
Feynman's own version of it -- he claims he couldn't quite follow Newton's -- is elementary in that it requires only high-school geometry and a vast amount of intelligence.
-- Wall Street Journal, 6/19/96, a review of Feynman's Lost Lecture

A computer, to print out a fact,
Will divide, multiply, and subtract.
But this output can be
No more than debris,
If the input was short of exact.

-- Gigo

God does not care about our mathematical difficulties. He integrates empirically.
Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.
-- Albert Einstein

Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.
-- Stephen Hawking

I used to think math was no fun,
'Cause I couldn't see how it was done.
Now Euler's my hero,
for I now see why 0
is e to the (i pi) plus 1.

Once upon a time, when I was training to be a mathematician, a group of us bright young students taking number theory discovered the names of the smaller prime numbers.
2: The Odd Prime --
It's the only even prime, therefore is odd. QED.
3: The True Prime --
Lewis Carroll: "If I tell you three times, it's true."
31: The Arbitrary Prime --
Determined by unanimous unvote. We needed an arbitrary prime in case the prof asked for one, and so had an election. 91 received the most votes (well, it looks prime) and 3+4i the next most. However, 31 was the only candidate to receive none at all.
Since the composite numbers are formed from primes, their qualities are derived from those primes. So, for instance, the number 6 is "odd but true", while the powers of 2 are all extremely odd numbers.

Q: How many mathematicians does it take to screw in a lightbulb?
A: None. It's left to the reader as an exercise.
A: Just one, once you've managed to present the problem in terms he/she is familiar with.

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell.
-- St. Augustine (354-430)
P.S. Augustine did really say that, but in his time there was no difference between mathematicians and astrologers. Astrologers told the future, which was deemed diabolic.

The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.
-- D'Alembert, Jean Le Rond (1717-1783)

There once was a student of Trinity,
Who tried to calculate the square root of infinity.
While counting the digits,
He was seized by the figits,
So he dropped math and took up divinity.

The tenants of pure mathematics are indistinguishable from theology.

University President: "Why is it that you physicists always require so much expensive equipment? Now the Department of Mathematics requires nothing but money for paper, pencils, and erasers . . . and the Department of Philosophy is better still. It doesn't even ask for erasers."
-- Told by Isaac Asimov

Young man, in mathematics you don't understand things, you just get used to them.
-- John von Neumann

Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.
-- Ernst Mach (1838-1916)

Mathematics: That branch of Human Thought which takes a finite set of trivial axioms and maps them to a countably infinite set of unintuitive theorems.

When we cannot use the compass of mathematics or the torch of experience... it is certain that we cannot take a single step forward.
-- Voltaire

Referee's report: This paper contains much that is new and much that is true. Unfortunately, that which is true is not new and that which is new is not true.

For the things of this world cannot be made known without a knowledge of mathematics.
-- Roger Bacon : Opus Majus part 4 Distinctia Prima cap 1, 1267.

Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure.
-- Martin Gardner

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.
-- Johannes Kepler (1571-1630)

Whoever despises the high wisdom of mathematics nourishes himself on delusion and will never still the sophistic sciences whose only product is an eternal uproar.
-- Leonardo da Vinci (1452-1519), quoted in "Mathematical Maxims and Minims", N. Rose, Raleigh NC:Rome Press Inc., 1988.

The biologist can push it back to the original protist, and the chemist can push it back to the crystal, but none of them touch the real question of why or how the thing began at all. The astronomer goes back untold million of years and ends in gas and emptiness, and then the mathematician sweeps the whole cosmos into unreality and leaves one with mind as the only thing of which we have any immediate apprehension. Cogito ergo sum, ergo omnia esse videntur. All this bother, and we are no further than Descartes. Have you noticed that the astronomers and mathematicians are much the most cheerful people of the lot? I suppose that perpetually contemplating things on so vast a scale makes them feel either that it doesn't matter a hoot anyway, or that anything so large and elaborate must have some sense in it somewhere.
-- Dorothy L. Sayers and/or R. Eustace.

I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it.
-- Malcom X

"Obvious" is the most dangerous word in mathematics.
-- Eric Temple Bell, 1883-1960

A statistician is a person who draws a mathematically precise line from an unwarranted assumption to a foregone conclusion. -- Anon