Q: What’s purple and commutes?
A: An abelian grape.

Q: What’s purple, commutes, and is worshipped by
a limited number of people?
A: A finitely-venerated abelian grape.

Q: What is lavender and commutes?
A: An abelian semigrape.

Q: What is purple and all of its offspring have been
committed to institutions?
A: A simple grape: it has no normal subgrapes.

Q: What’s purple, round, and doesn’t get much for
Christmas?
A: A finitely presented grape.

Q: What’s an abelian group under addition, closed,
associative, distributive, and bears a curse?
A: The Ring of the Nibelung.

Q: What’s nutritious and commutes?
A: An abelian soup.

Q: What’s hot, chunky, and acts on a polygon?
A: Dihedral soup.

Q: What’s sour, yellow, and equivalent to the Axiom
of Choice?
A: Zorn’s Lemon.

Q: What is brown, furry, runs to the sea, and is equivalent
to the Axiom of Choice?
A: Zorn’s lemming.

Q: What is green and homeomorphic to the open unit
interval?
A: The real lime.

Q: What’s yellow, linear, normed, and complete?
A: A Bananach space.

Q: What do you call a young eigensheep?
A: A lamb, duh!

Q: What’s the value of a contour integral around
Western Europe?
A: Zero, because all the Poles are in Eastern Europe.
Addendum: Actually, there ARE some Poles in Western
Europe, but they are removable! 2

Q: Why did the mathematician name his dog
“Cauchy”?
A: Because he left a residue at every pole.

Q: What is a topologist?
A: Someone who cannot distinguish between a
doughnut and a coffee cup.

Q: Why didn’t Newton discover group theory?
A: Because he wasn’t Abel.

Q: What do you get if you cross an elephant and a
banana?
A: |elephant|*|banana|*sin(theta).

Q: What do you get if you cross a mosquito with a
mountain climber?
A: You can’t cross a vector with a scalar.
or this variation

Q: What do you get when you cross a mountain goat
and a mountain climber?
A: Nothing—you can’t cross two scalars.3

Q: What is a compact city?
A: It’s a city that can be guarded by finitely many
nearsighted policemen.4

Q: What’s a dilemma?
A: A lemma that produces two results.

Q: What’s a polar bear?
A: A rectangular bear after a coordinate transform.

Q: What goes “Pieces of seven! Pieces of seven!”
A: A parroty error.

Q: Why can’t you grow wheat in Z/6Z?
A: It’s not a field.

Q: What’s big, grey, and proves the uncountability
of the reals?
A: Cantor’s diagonal elephant.5

Q: What is gray and huge and has integer coefficients?
A: An elephantine equation.

Q: What is very old, used by farmers, and obeys the
fundamental theorem of arithmetic?
A: An antique tractorization domain.

Q: What is hallucinogenic and exists for every group
with order divisible by pk?
A: A psilocybin p-subgroup.

Q: What is often used by Canadians to help solve certain
differential equations?
A: The Lacross transform.

Q: What is clear and used by trendy sophisticated
engineers to solve other differential equations?
A: The Perrier transform.

Q: Who knows everything there is to be known about
vector analysis?
A: The Oracle of del phi!

Q: Why can fish from the United States enter Canadian
waters without a passport?
A: This is permitted by the Law of Aquatic reciprocity!

Q: Why are topologists especially prone to malaria?
A: This disease comes from the Tietze fly!! 6

Q: Why do truncated Maclaurin series fit the original
function so well?
A: Because they are “Taylor” made.

Q: What is locally like a ring and very evil?
A: A devilish scheme.

Q: What is a proof?
A: One-half percent of alcohol.

Q: Can you prove Lagrange’s Identity?
A: Are you kidding? It’s really hard to prove the identity
of someone who’s been dead for over 150 years!

Q: What is black and white ivory and fills space?
A: A piano curve.

Q: What’s polite and works for the phone company?
A: A deferential operator.

Q: What does an analytic number theorist say when
he is drowning?
A: Log-log, log-log, log-log, ….

Q: What does a topologist call a virgin?
A: Simply connected.7

Q: How many topologists does it take to change a
lightbulb?
A: Just one, but what will you do with the doughnut?

Q: How many number theorists does it take to
change a lightbulb?
A: This is not known, but it is conjectured to be an
elegant prime.

Q: How many geometers does it take to screw in a
lightbulb?
A: None. You can’t do it with a straightedge and compass.

Q: How many analysts does it take to screw in a lightbulb?
A: Three. One to prove existence, one to prove
uniqueness, and one to derive a nonconstructive
algorithm to do it.

Q: How many Bourbakists does it take to replace a
lightbulb?
A: Changing a lightbulb is a special case of a more
general theorem concerning the maintenance and
repair of an electrical system. To establish upper and
lower bounds for the number of personnel required,
we must determine whether the sufficient conditions
of Lemma 2.1 (Availability of personnel) and those
of Corollary 2.3.55 (Motivation of personnel) apply.
If and only if these conditions are met, we derive the
result by an application of the theorems in Section
3.1123. The resulting upper bound is, of course, a
result in an abstract measure space, in the
weak-* topology.

Q: How many mathematicians does it take to screw
in a lightbulb?
A: 0.999999….

Q: How many lightbulbs does it take to change a
lightbulb?
A: One, if it knows its own Gödel number.

Q: Why did the chicken cross the road?
A: Gödel: It cannot be proved whether the chicken
crossed the road.
Some other answers to this question provided
by famous mathematicians include:

Q: Why did the chicken cross the road?
A: Erdös: It was forced to do so by the chicken-hole
principle.

Q: Why did the chicken cross the road?
A: Riemann: The answer appears in Dirichlet’s lectures.

Q: Why did the chicken cross the road?
A: Fermat: It did not fit on the margin on this side.

Q: Why did the chicken cross the Möbius strip?
A: To get to the other–er….
If you cannot prove the theorem, prove a different
one:
Q: Why did the chicken cross the Möbius strip?
A: To get to the same side.


“Top Ten
Excuses for Not Doing Homework”:
• I accidentally divided by zero and my paper burst
into flames.
• Isaac Newton’s birthday.
• 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 fourdimensional
dog got in and ate it.
• I couldn’t figure out whether i am the square of
negative one or i is the square root of negative
one.
• I took time out to snack on a doughnut and a cup
of coffee [and] I spent the rest of the night trying
to figure which one to dunk.
• I could have sworn I put the home work inside a
Klein bottle, but this morning I couldn’t find it.



How to prove it. Guide for lecturers.

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 NPcomplete
[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 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 appeal to intuition:
Cloud-shaped drawings frequently help here.



What is the difference between an argument
and a proof? An argument will
convince a reasonable man, but a proof
is needed to convince an unreasonable
one.




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

Theorem. All dogs have nine legs.
Proof. Would you agree that no dog has five
legs? Would you agree that a dog has four legs
more than no dog? 4 + 5 =?

Theorem. All positive integers are interesting.
Proof. Assume the contrary. Then there is a lowest
noninteresting positive integer. But, hey, that’s
pretty interesting! A contradiction.

Theorem. 3 = 4.
Proof. Suppose
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.

You know that during the Great Flood,
Noah brought along two of every species
for reproductive purposes. Well, after a
few weeks on the ark, all the couples
were getting along fine, except for these
two snakes. Day and night, Noah worried
that this was going to mean the end
of this species. Finally when the flood
ended and the ark hit ground, the two
snakes darted out of the ship and headed
to the nearest picnic table where they
started to “go at it”. It was then that
Noah realized that…Adders can’t multiply
without their log tables.

Math song:

N bottles of beer on the wall,
N bottles of beer,
You take one down, and pass it around,
N − 1 bottles of beer on the wall.

The song begins at N = 100 and then repeats until
there are no bottles of beer remaining. It is often
sung by children on long car rides, as a way of keeping
them busy. The following mathematical variant
is an amusing way to keep the children occupied
a bit longer:

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer,
You take one down, and pass it
around,
Aleph-null bottles of beer on the wall.


Engineers think that equations approximate
the real world. Physicists think
that the real world approximates equations.
Mathematicians are unable to
make the connection.


An engineer, a physicist, and a mathematician
are staying in a hotel. The engineer
wakes up and smells smoke. He
goes out into the hallway and sees a fire,
so he fills a trash can from his room
with water and douses the fire. He goes
back to bed. Later, the physicist wakes
up and smells the smoke. He opens his
door and sees a fire in the hallway. He
walks down the hall to a fire hose and
after calculating the flame velocity, distance,
water pressure, trajectory, and
so forth, extinguishes the fire with the
minimum amount of water and energy
needed. Later the mathematician wakes
up and smells smoke. He goes to the hall,
sees the fire and then the fire hose. He
thinks for a moment and then exclaims,
“Ah, a solution exists!” and then goes
back to bed.

A physicist and a mathematician are
sitting in a faculty lounge. Suddenly, the
coffee machine catches on fire. The
physicist grabs a bucket and leaps toward
the sink, fills the bucket with water,
and puts out the fire. Second day, the
same two sit in the same lounge. Again
the coffee machine catches on fire. This
time, the mathematician stands up, gets
a bucket, and hands the bucket to the
physicist, thus reducing the problem to
a previously solved one.

A mathematician and an engineer are
on a desert island. They find two palm
trees with one coconut each. The engineer
shinnies up one tree, gets the coconut,
and eats it. The mathematician
shinnies up the other tree, gets the coconut,
climbs the other tree and puts it
there. “Now we’ve reduced it to a problem
we know how to solve.”

There are a mathematician and a physicist
and a burning building with people
inside. There are a fire hydrant and a
hose on the sidewalk. The physicist has
to put the fire out…so, he attaches the
hose to the hydrant, puts the fire out, and
saves the house and the family. Then
they put the people back in the house, set
it on fire, and ask the mathematician
to solve the problem. So, he takes the
hose off the hydrant and lays it on the
sidewalk. “Now I’ve reduced it to a
previously solved problem,” and walks
away.


One day a farmer called up an engineer,
a physicist, and a mathematician
and asked them to fence in the largest
possible area with the least amount of
fence. The engineer made the fence in
a circle and proclaimed that he had the
most efficient design. The physicist made
a long, straight line and proclaimed “We
can assume the length is infinite…” and
pointed out that fencing off half of the
Earth was certainly a more efficient way
to do it. The mathematician just laughed
at them. He built a tiny fence around
himself and said, “I declare myself to be
on the outside.”

A biologist, a physicist, and a mathematician
were sitting in a street café
watching the crowd. Across the street
they saw a man and a woman entering
a building. Ten minutes later they reappeared
together with a third person.
“They have multiplied,” said the biologist.
“Oh no, an error in measurement,” the
physicist sighed. “If exactly one person
enters the building now, it will be empty
again,” the mathematician concluded.

Three men with degrees in mathematics,
physics, and biology are locked up in dark
rooms for research reasons. A week later
the researchers open the door and the biologist
steps out and reports: “Well, I sat
around until I started to get bored, then
I searched the room and found a tin which
I smashed on the floor. There was food
in it which I ate when I got hungry. That’s
it.” Then they free the man with the degree
in physics and he says: “I walked
along the walls to get an image of the
room’s geometry, then I searched it. There
was a metal cylinder at five feet into the
room and two feet left of the door. It felt
like a tin and I threw it at the left wall at
the right angle and velocity for it to crack
open.” Finally, the researchers open the
third door and hear a faint voice out of
the darkness: “Let C be an open can.”


There was a mad scientist (a mad …
social…scientist) who kidnapped three
colleagues, an engineer, a physicist, and
a mathematician, and locked each of
them in separate cells with plenty of
canned food and water but no can
opener. A month later, returning, the
mad scientist went to the engineer’s cell
and found it long empty. The engineer
had constructed a can opener from
pocket trash, used aluminum shavings
and dried sugar to make an explosive,
and escaped. The physicist had worked
out the angle necessary to knock the lids
off the tin cans by throwing them against
the wall. She was developing a good
pitching arm and a new quantum theory.
The mathematician had stacked the
unopened cans into a surprising solution
to the kissing problem; his desiccated
corpse was propped calmly against a
wall, and this was inscribed on the floor
in blood: Theorem: If I can’t open these
cans, I’ll die. Proof: Assume the opposite.…


A mathematician, a physicist, and an
engineer were traveling through Scotland
when they saw a black sheep
through the window of the train. “Aha,”
says the engineer, “I see that Scottish
sheep are black.” “Hmm,” says the physicist,
“You mean that some Scottish sheep
are black.” “No,” says the mathematician,
“All we know is that there is at
least one sheep in Scotland, and that at
least one side of that one sheep is black!”

Three men are in a hot-air balloon. Soon,
they find themselves lost in a canyon
somewhere. One of the three men says,
“I’ve got an idea. We can call for help in
this canyon and the echo will carry our
voices far.” So he leans over the basket
and yells out, “Helloooooo! Where are
we?” (They hear the echo several times.)
Fifteen minutes later, they hear this echoing
voice: “Hellooooo! You’re lost!!” One
of the men says, “That must have been
a mathematician.” Puzzled, one of the
other men asks, “Why do you say that?”
The reply: “For three reasons: (1) He took
a long time to answer, (2) he was
absolutely correct, and (3) his answer
was absolutely useless.”


An engineer, a physicist, and a mathematician
find themselves in an anecdote,
indeed an anecdote quite similar to
many that you have no doubt already
heard. After some observations and
rough calculations the engineer realizes
the situation and starts laughing. A few
minutes later the physicist understands
too and chuckles to himself happily, as
he now has enough experimental evidence
to publish a paper. This leaves
the mathematician somewhat perplexed,
as he had observed right away that he
was the subject of an anecdote and deduced
quite rapidly the presence of
humor from similar anecdotes, but considers
this anecdote to be too trivial a
corollary to be significant, let alone
funny.


A mathematics professor was lecturing
to a class of students. As he wrote something
on the board, he said to the class
“Of course, this is immediately obvious.”
Upon seeing the blank stares of the students,
he turned back to contemplate
what he had just written. He began to
pace back and forth, deep in thought.
After about 10 minutes, just as the silence
was beginning to become uncomfortable,
he brightened, turned to the
class and said, “Yes, it IS obvious.”



Some of the best-known examples of mathematical
folk humor consist of anecdotes, often apocryphal,
attached to specific famous individuals.
Not infrequently the protagonist is an absent-minded
professor. In theory, it could be a professor from
any academic discipline, but more often than not, it
is a mathematician. One such classic text has Norbert
Wiener encountering a student on the street and engaging
him in theoretical discussion. At the end of
the interaction, the student is startled to hear the
professor ask, “Which way was I headed when we
met?” The student points, saying, “You were going
that way, sir.” “Good,” says Wiener, “Then I’ve had my
lunch.” But probably the best-known Wiener absentminded-
professor anecdote goes as follows:

One day the Wiener family was scheduled
to move into a new house. Mrs. Wiener,
mindful of her husband’s propensity for
forgetting, wrote the new address on a
slip of paper and handed it to him. He
scoffed, saying, “I wouldn’t forget such
an important thing,” but he took the slip
of paper and put it in his pocket. Later
that same day at the university a colleague
came by his office with an interesting
problem. Wiener searched for a
piece of paper and took the slip from his
pocket to use to write some mathematical
equations. When he finished, he crumpled
up the slip of paper and threw it
away. That evening, he remembered
there was something about a new house
but he couldn’t find the slip of paper
with the address on it. Without any alternative
course of action, he returned to
his old home, where he spotted a little girl
on the sidewalk. “Say, little girl,” he said,
“Do you know where the Wieners live?”
The girl replied, “That’s o.k., Daddy,
Mommy sent me to get you.”

There are apparently no limits on what an
absent-minded professor might forget, as attested
by the following lesser-known Wiener anecdote [K]:

One day, a student saw Wiener in the
post office and wanted to introduce himself
to the famous professor. After all,
how many M.I.T. students could say that
they had actually shaken the hand of
Norbert Wiener? However, the student
wasn’t sure how to approach the man.
The problem was aggravated by the fact
that Wiener was pacing back and forth,
deeply lost in thought. Were the student
to interrupt Wiener, who knows what
profound idea might be lost? Still, the student
screwed up his courage and
approached the great man. “Good morning,
Professor Wiener,” he said. The professor
looked up, struck his forehead,
and said “That’s it: Wiener!”




Two mathematicians are in a bar. The
first one says to the second that the average
person knows very little about
basic mathematics. The second one disagrees
and claims that most people can
cope with a reasonable amount of math.
The first mathematician goes off to the
washroom, and in his absence the second
calls over the waitress. He tells her
that in a few minutes, after his friend has
returned, he will call her over and ask
her a question. All she has to do is answer
“one third x cubed.” She repeats
“one thir–dex cue?” He repeats “one third
x cubed.” She asks, “one thir dex cuebd?”
“Yes, that’s right,” he says. So she agrees,
and goes off mumbling to herself, “one
thir dex cuebd…”. The first guy returns
and the second proposes a bet to prove
his point, that most people do know
something about basic math. He says
he will ask the blonde waitress an integral,
and the first laughingly agrees.
The second man calls over the waitress
and asks “what is the integral of x
squared?” The waitress says “one third
x cubed” and while walking away, turns
back and says over her shoulder, “plus
a constant!”


At New York’s Kennedy Airport today, an
individual was arrested trying to board
a flight while in possession of a ruler, a
protractor, a setsquare, a slide rule, and
a calculator. At a morning press conference,
Attorney General John Ashcroft
said he believes the man is a member of
the notorious al-gebra movement.9 He is
being charged by the FBI with carrying
weapons of math instruction.
“Al-gebra is a fearsome cult,” Ashcroft
said. “They desire average solutions by
means and extremes, and sometimes go
off on tangents in a search of absolute
value. They use secret code names like
‘x’ and ‘y’ and refer to themselves as
‘unknowns’, but we have determined
they belong to a common denominator
of the axis with coordinates in every
country. As the Greek philanderer Isosceles
used to say, there are three sides to
every triangle,” Ashcroft declared.
When asked to comment on the arrest,
President Bush said, “If God had wanted
us to have better weapons of math instruction,
He would have given us more
fingers and toes. I am gratified that our
government has given us a sine that it
is intent on protracting us from those
who are willing to disintegrate us with
calculus disregard. Under the circumferences,
we must differentiate their
root, make our point, and draw the line.”
President Bush warned, “These weapons
of math instruction have the potential
to decimal everything in their math on
a scalene never before seen unless we
become exponents of a higher power
and begin to factor in random facts of
vertex.”
Attorney General Ashcroft said, “Read
my ellipse. Their days are numbered as
the hypotenuse tightens around their
necks.”