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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, "Helllloooooo! Where are
we?" (They hear the echo several times.)
15 minutes later, they hear this echoing voice: "Helllloooooo! 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."
-------------------------------------------------------------------------------
Actually, I prefer the IBM version of this joke...
A small, 14-seat plane is circling for a landing in Atlanta. It's
totally fogged in, zero visibility, and suddenly there's a small
electrical fire in the cockpit which disables all of the instruments
and the radio. The pilot continues circling, totally lost, when
suddenly he finds himself flying next to a tall office building.
He rolls down the window (this particular airplane happens to have
roll-down windows) and yells to a person inside the building, "Where
are we?"
The person responds "In an airplane!"
The pilot then banks sharply to the right, circles twice, and makes a
perfect landing at Atlanta International.
As the passengers emerge, shaken but unhurt, one of them says to the
pilot, "I'm certainly glad you were able to land safely, but I don't
understand how the response you got was any use."
"Simple," responded the pilot. "I got an answer that was completely
accurate and totally irrelevant to my problem, so I knew it had to be
the IBM building."
-------------------------------------------------------------------------------
(I'm not sure if the following one is a true story or not)
The great logician Bertrand Russell (or was it A.N. Whitehead?)
once claimed that he could prove anything if given that 1+1=1.
So one day, some smarty-pants asked him, "Ok. Prove that
you're the Pope."
He thought for a while and proclaimed, "I am one. The Pope
is one. Therefore, the Pope and I are one."
[NOTE: The following is from merritt@Gendev.slc.paramax.com (Merritt).
The story about 1+1=1 causing ridiculous consequences was, I believe,
originally the product of a conversation at the Trinity High Table.
It is recorded in Sir Harold Jeffreys' Scientific Inference, in a note
to chapter one. Jeffreys remarks that the fact that everything
followed from a single contradiction had been noticed by Aristotle (I
doubt this way of putting it is quite correct, but that is beside the
point). He goes on to say that McTaggart denied the consequence: "If
2+2=5, how can you prove that I am the pope?" Hardy is supposed to
have replied: "If 2+2=5, 4=5; subtract 3; then 1=2; but McTaggart and
the pope are two; therefore McTaggart and the pope are one." When I
consider this story, I am astonished at how much more brilliant some
people are than I (quite independent of the fallacies in the
argument).
Since McTaggart, Hardy, Whitehead, and Russell (the last two of whom
were credited with a variant of Hardy's argument in your post) were
all fellows of Trinity and Jeffreys (their exact contemporary) was a
fellow of St. Johns, I suspect that (whatever the truth of Jeffreys'
story) it is very unlikely that Whitehead or Russell had anything to do
with it. The extraordinary point to me about the story is that Hardy
was able to snap this argument out between mouthfuls, so to speak, and
he was not even a logician at all. This is probably why it came in
some people's minds to be attributed to one or other of the famous
Trinity logicians.
-------------------------------------------------------------------------------
THE STORY OF BABEL:
In the beginning there was only one kind of Mathematician: the Topologist.
And they grew to large numbers and prospered.
One day they looked up in the heavens and desired to reach up as far
as the eye could see. So they set out in building a Mathematical
edifice that was to reach up as far as "up" went. Further and further
up they went ... until one night the edifice collapsed under the
weight of paradox.
The following morning saw only rubble where there once was a huge
structure reaching to the heavens. One by one, the Mathematicians
climbed out from under the rubble. It was a miracle that nobody was
killed; but when they began to speak to one another, SUPRISE of all
surprises! they could not understand each other. They all spoke
different languages. They all fought amongst themselves and each went
about their own way. To this day the Topologists remain the original
Mathematicians.
- adapted from an American Indian legend
of the Mound Of Babel
-------------------------------------------------------------------------------
Methods of Mathematical Proof
This is from _A Random Walk in Science_ (by Joel E. Cohen?):
To illustrate the various methods of proof we give an example of a
logical system.
THE PEJORATIVE CALCULUS
Lemma 1. All horses are the same colour.
(Proof by induction)
Proof. It is obvious that one horse is the same colour. Let us assume
the proposition P(k) that k horses are the same colour and use this to
imply that k+1 horses are the same colour. Given the set of k+1 horses,
we remove one horse; then the remaining k horses are the same colour,
by hypothesis. We remove another horse and replace the first; the k
horses, by hypothesis, are again the same colour. We repeat this until
by exhaustion the k+1 sets of k horses have been shown to be the same
colour. It follows that since every horse is the same colour as every
other horse, P(k) entails P(k+1). But since we have shown P(1) to be
true, P is true for all succeeding values of k, that is, all horses are
the same colour.
Theorem 1. Every horse has an infinite number of legs.
(Proof by intimidation.)
Proof. 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 cer-
tainly an odd number of legs for a horse. But the only number that is
both odd and even is infinity. Therefore horses have an infinite num-
ber of legs. Now to show that this is general, suppose that somewhere
there is a horse with a finite number of legs. But that is a horse of
another colour, and by the lemma that does not exist.
Corollary 1. Everything is the same colour.
Proof. The proof of lemma 1 does not depend at all on the nature of the
object under consideration. The predicate of the antecedent of the uni-
versally-quantified conditional 'For all x, if x is a horse, then x is
the same colour,' namely 'is a horse' may be generalized to 'is anything'
without affecting the validity of the proof; hence, 'for all x, if x is
anything, x is the same colour.'
Corollary 2. Everything is white.
Proof. If a sentential formula in x is logically true, then any parti-
cular substitution instance of it is a true sentence. In particular
then: 'for all x, if x is an elephant, then x is the same colour' is
true. Now it is manifestly axiomatic that white elephants exist (for
proof by blatant assertion consult Mark Twain 'The Stolen White Ele-
phant'). Therefore all elephants are white. By corollary 1 everything
is white.
Theorem 2. Alexander the Great did not exist and he had an infinite
number of limbs.
Proof. We prove this theorem in two parts. First we note the obvious
fact that historians always tell the truth (for historians always take
a stand, and therefore they cannot lie). Hence we have the historically
true sentence, 'If Alexander the Great existed, then he rode a black
horse Bucephalus.' But we know by corollary 2 everything is white;
hence Alexander could not have ridden a black horse. Since the conse-
quent of the conditional is false, in order for the whole statement to
be true the antecedent must be false. Hence Alexander the Great did not
exist.
We have also the historically true statement that Alexander was warned
by an oracle that he would meet death if he crossed a certain river. He
had two legs; and 'forewarned is four-armed.' This gives him six limbs,
an even number, which is certainly an odd number of limbs for a man.
Now the only number which is even and odd is infinity; hence Alexander
had an infinite number of limbs. We have thus proved that Alexander the
Great did not exist and that he had an infinite number of limbs.
-------------------------------------------------------------------------------
Several students were asked the following problem:
Prove that all odd integers are prime.
Well, the first student to try to do this was a math student. Hey
says "Hmmm... Well, 1 is prime, 3 is prime, 5 is prime, and by
induction, we have that all the odd integers are prime."
Of course, there are some jeers from some of his friends. The physics
student then said, "I'm not sure of the validity of your proof, but I
think I'll try to prove it by experiment." He continues, "Well, 1 is
prime, 3 is prime, 5 is prime, 7 is prime, 9 is ... uh, 9 is an
experimental error, 11 is prime, 13 is prime... Well, it seems that
you're right."
The third student to try it was the engineering student, who
responded, "Well, actually, I'm not sure of your answer either. Let's
see... 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is ..., 9 is
, well if you approximate, 9 is prime, 11 is prime, 13 is prime...
Well, it does seem right."
Not to be outdone, the computer science student comes along and says
"Well, you two sort've got the right idea, but you'd end up taking too
long doing it. I've just whipped up a program to REALLY go and prove
it..." He goes over to his terminal and runs his program. Reading
the output on the screen he says, "1 is prime, 1 is prime, 1 is prime,
1 is prime...."
-------------------------------------------------------------------------------
Mathematician: 3 is a prime, 5 is a prime, 7 is a prime,
9 is not a prime - counter-example - claim is false.
Physicist: 3 is a prime, 5 is a prime, 7 is a prime,
9 is an experimental error, 11 is a prime, ...
Engineer: 3 is a prime, 5 is a prime, 7 is a prime,
9 is a prime, 11 is a prime, ...
Computer scientist: 3's a prime, 5's a prime, 7's a prime, 7's a prime,
7's a prime, ...
Computer scientist using Unix: 3's a prime, 5's a prime, 7's a prime,
segmentation fault
They all overlooked that even 2's a prime!!
I figure that 2 is the oddest prime of all, because it's the
only one that's even!
-------------------------------------------------------------------------------
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.
-------------------------------------------------------------------------------
My geometry teacher was sometimes acute, and sometimes
obtuse, but always, he was right.
-------------------------------------------------------------------------------
And now, for some really bad picture jokes (that I heard at Cal Poly SLO) :
Q: What's the title of this picture ?
.. .. ____ .. ..
\\===/====\==\\==
|| | | ||
|| |____| ||
|| | | ||
|| \____/ ||
|| ||
|| ||
|| ||
|| ||
|| ||
|| ||
|| ||
|| ||
|| ||
|| (\ ||
|| ) ) ||
|| //||\\ ||
A: Hypotenuse
-------
Q: What quantity is represented by this ?
/\ /\ /\
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/______\ /______\ /______\
|| || ||
|| || ||
A: 9, tree + tree + tree
Q: A dust storm blows through, now how much do you have ?
A: 99, dirty tree + dirty tree + dirty tree
-------------------------------------------------------------------------------
I saw the following scrawled on a math office blackboard in college:
1 + 1 = 3, for large values of 1
-------------------------------------------------------------------------------
lim ----
8->9 \/ 8 = 3
Along the same lines:
lim sqrt (3) = 2
3->4
-------------------------------------------------------------------------------
Asked how his pet parrot died, the mathematician answered
"Polynomial. Polygon."
-------------------------------------------------------------------------------
Lumberjacks make good musicians because of their natural logarithms.
-------------------------------------------------------------------------------
Q: What is Quayle-o-phobia?
A: The fear of natural logarithms.
(Hint: Quayle and the letter "e" made news.)
-------------------------------------------------------------------------------
Pie are not square. Pie are round. Cornbread are square.
-------------------------------------------------------------------------------
"The integral of e to the x is equal to f of the quantity
u to the n."
/ x n
| e = f(u )
/
-------------------------------------------------------------------------------
A physics joke:
"Energy equals milk chocolate square"
-------------------------------------------------------------------------------
Russell to Whitehead: "My Godel is killing me!"
-------------------------------------------------------------------------------
Von Neumann and Norbert Wiener were both the subject of many dotty
professor stories. Von Neumann supposedly had the habit of simply
writing answers to homework assignments on the board (the method of
solution being, of course, obvious) when he was asked how to solve
problems. One time one of his students tried to get more helpful
information by asking if there was another way to solve the problem.
Von Neumann looked blank for a moment, thought, and then answered,
"Yes".
Wiener was in fact very absent minded. The following story is told
about him: When they moved from Cambridge to Newton his wife, knowing
that he would be absolutely useless on the move, packed him off to MIT
while she directed the move. Since she was certain that he would
forget that they had moved and where they had moved to, she wrote down
the new address on a piece of paper, and gave it to him. Naturally,
in the course of the day, an insight occurred to him. He reached in
his pocket, found a piece of paper on which he furiously scribbled
some notes, thought it over, decided there was a fallacy in his idea,
and threw the piece of paper away. At the end of the day he went home
(to the old address in Cambridge, of course). When he got there he
realized that they had moved, that he had no idea where they had moved
to, and that the piece of paper with the address was long gone.
Fortunately inspiration struck. There was a young girl on the street
and he conceived the idea of asking her where he had moved to, saying,
"Excuse me, perhaps you know me. I'm Norbert Wiener and we've just
moved. Would you know where we've moved to?" To which the young girl
replied, "Yes daddy, mommy thought you would forget."
The capper to the story is that I asked his daughter (the girl in the
story) about the truth of the story, many years later. She said that
it wasn't quite true -- that he never forgot who his children were!
The rest of it, however, was pretty close to what actually happened...
-------------------------------------------------------------------------------
The USDA once wanted to make cows produce milk faster, to improve the
dairy industry.
So, they decided to consult the foremost biologists and recombinant
DNA technicians to build them a better cow. They assembled this team
of great scientists, and gave them unlimited funding. They requested
rare chemicals, weird bacteria, tons of quarantine equipment, there
was a horrible typhus epidemic they started by accident, and, 2 years
later, they came back with the "new, improved cow." It had a milk
production improvement of 2% over the original.
They then tried with the greatest Nobel Prize winning chemists around.
They worked for six months, and, after requisitioning tons of chemical
equipment, and poisoning half the small town in Colorado where they
were working with a toxic cloud from one of their experiments, they
got a 5% improvement in milk output.
The physicists tried for a year, and, after ten thousand cows were
subjected to radiation therapy, they got a 1% improvement in output.
Finally, in desperation, they turned to the mathematicians. The
foremost mathematician of his time offered to help them with the
problem. Upon hearing the problem, he told the delegation that they
could come back in the morning and he would have solved the problem.
In the morning, they came back, and he handed them a piece of paper
with the computations for the new, 300% improved milk cow.
The plans began:
"A Proof of the Attainability of Increased Milk Output from Bovines:
Consider a spherical cow......"
-------------------------------------------------------------------------------
An engineer, a mathematician, and a physicist went to the races one
Saturday and laid their money down. Commiserating in the bar after
the race, the engineer says, "I don't understand why I lost all my
money. I measured all the horses and calculated their strength and
mechanical advantage and figured out how fast they could run..."
The physicist interrupted him: "...but you didn't take individual
variations into account. I did a statistical analysis of their
previous performances and bet on the horses with the highest
probability of winning..."
"...so if you're so hot why are you broke?" asked the engineer. But
before the argument can grow, the mathematician takes out his pipe and
they get a glimpse of his well-fattened wallet. Obviously here was a
man who knows something about horses. They both demanded to know his
secret.
"Well," he says, between puffs on the pipe, "first I assumed all the
horses were identical and spherical..."
-------------------------------------------------------------------------------
Theorem : All positive integers are equal.
Proof : Sufficient to show that for any two positive integers, A and B,
A = B. Further, it is sufficient to show that for all N > 0, if A
and B (positive integers) satisfy (MAX(A, B) = N) then A = B.
Proceed by induction.
If N = 1, then A and B, being positive integers, must both be 1.
So A = B.
Assume that the theorem is true for some value k. Take A and B
with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence
(A-1) = (B-1). Consequently, A = B.
-------------------------------------------------------------------------------
A bunch of Polish scientists decided to flee their repressive
government by hijacking an airliner and forcing the pilot to fly them
to a western country. They drove to the airport, forced their way on
board a large passenger jet, and found there was no pilot on board.
Terrified, they listened as the sirens got louder. Finally, one of
the scientists suggested that since he was an experimentalist, he
would try to fly the aircraft.
He sat down at the controls and tried to figure them out. The sirens
got louder and louder. Armed men surrounded the jet. The would be
pilot's friends cried out, "Please, please take off now!!!
Hurry!!!!!!"
The experimentalist calmly replied, "Have patience. I'm just a simple
pole in a complex plane."
-------------------------------------------------------------------------------
A group of Polish tourists is flying on a small airplane through the
Grand Canyon on a sightseeing tour. The tour guide announces: "On the
right of the airplane, you can see the famous Bright Angle Falls."
The tourists leap out of their seats and crowd to the windows on the
right side. This causes a dynamic imbalance, and the plane violently
rolls to the side and crashes into the canyon wall. All aboard are
lost. The moral to this episode is: always keep your poles off the
right side of the plane.
Caveat: While this joke mentions Polish people, it is not, in my
opinion, in the category of the infamous Polish jokes. I hope no one
is offended but only humored.
-------------------------------------------------------------------------------
Hiawatha Designs an Experiment
Hiawatha, mighty hunter,
He could shoot ten arrows upward,
Shoot them with such strength and swiftness
That the last had left the bow-string
Ere the first to earth descended.
This was commonly regarded
As a feat of skill and cunning.
Several sarcastic spirits
Pointed out to him, however,
That it might be much more useful
If he sometimes hit the target.
"Why not shoot a little straighter
And employ a smaller sample?"
Hiawatha, who at college
Majored in applied statistics,
Consequently felt entitled
To instruct his fellow man
In any subject whatsoever,
Waxed exceedingly indignant,
Talked about the law of errors,
Talked about truncated normals,
Talked of loss of information,
Talked about his lack of bias,
Pointed out that (in the long run)
Independent observations,
Even though they missed the target,
Had an average point of impact
Very near the spot he aimed at,
With the possible exception
of a set of measure zero.
"This," they said, "was rather doubtful;
Anyway it didn't matter.
What resulted in the long run:
Either he must hit the target
Much more often than at present,
Or himself would have to pay for
All the arrows he had wasted."
Hiawatha, in a temper,
Quoted parts of R. A. Fisher,
Quoted Yates and quoted Finney,
Quoted reams of Oscar Kempthorne,
Quoted Anderson and Bancroft
(practically in extenso)
Trying to impress upon them
That what actually mattered
Was to estimate the error.
Several of them admitted:
"Such a thing might have its uses;
Still," they said, "he would do better
If he shot a little straighter."
Hiawatha, to convince them,
Organized a shooting contest.
Laid out in the proper manner
Of designs experimental
Recommended in the textbooks,
Mainly used for tasting tea
(but sometimes used in other cases)
Used factorial arrangements
And the theory of Galois,
Got a nicely balanced layout
And successfully confounded
Second order interactions.
All the other tribal marksmen,
Ignorant benighted creatures
Of experimental setups,
Used their time of preparation
Putting in a lot of practice
Merely shooting at the target.
Thus it happened in the contest
That their scores were most impressive
With one solitary exception.
This, I hate to have to say it,
Was the score of Hiawatha,
Who as usual shot his arrows,
Shot them with great strength and swiftness,
Managing to be unbiased,
Not however with a salvo
Managing to hit the target.
"There!" they said to Hiawatha,
"That is what we all expected."
Hiawatha, nothing daunted,
Called for pen and called for paper.
But analysis of variance
Finally produced the figures
Showing beyond all peradventure,
Everybody else was biased.
And the variance components
Did not differ from each other's,
Or from Hiawatha's.
(This last point it might be mentioned,
Would have been much more convincing
If he hadn't been compelled to
Estimate his own components
From experimental plots on
Which the values all were missing.)
Still they couldn't understand it,
So they couldn't raise objections.
(Which is what so often happens
with analysis of variance.)
All the same his fellow tribesmen,
Ignorant benighted heathens,
Took away his bow and arrows,
Said that though my Hiawatha
Was a brilliant statistician,
He was useless as a bowman.
As for variance components
Several of the more outspoken
Make primeval observations
Hurtful of the finer feelings
Even of the statistician.
In a corner of the forest
Sits alone my Hiawatha
Permanently cogitating
On the normal law of errors.
Wondering in idle moments
If perhaps increased precision
Might perhaps be sometimes better
Even at the cost of bias,
If one could thereby now and then
Register upon a target.
W. E. Mientka, "Professor Leo Moser -- Reflections of a Visit"
American Mathematical Monthly, Vol. 79, Number 6 (June-July, 1972)
See also "Applied Dynamic Programming" by Bellman and Dreyfuss, prior to 1962.
-------------------------------------------------------------------------------
An assemblage of the most gifted minds in the world were all posed the
following question:
"What is 2 * 2 ?"
The engineer whips out his slide rule (so it's old) and shuffles it
back and forth, and finally announces "3.99".
The physicist consults his technical references, sets up the problem
on his computer, and announces "it lies between 3.98 and 4.02".
The mathematician cogitates for a while, oblivious to the rest of the
world, then announces: "I don't what the answer is, but I can tell
you, an answer exists!".
Philosopher: "But what do you _mean_ by 2 * 2 ?"
Logician: "Please define 2 * 2 more precisely."
Accountant: Closes all the doors and windows, looks around carefully,
then asks "What do you _want_ the answer to be?"
Computer Hacker: Breaks into the NSA super-computer and gives the answer.
-------------------------------------------------------------------------------
Economist: Someone who is good with numbers but lacks the personality
to be an accountant.
-------------------------------------------------------------------------------
Old mathematicians never die; they just lose some of their functions.
-------------------------------------------------------------------------------
During a class of calculus my lecturer suddenly checked himself and
stared intently at the table in front of him for a while. Then he
looked up at us and explained that he thought he had brought six piles
of papers with him, but "no matter how he counted" there was only five
on the table. Then he became silent for a while again and then told
the following story:
"When I was young in Poland I met the great mathematician Waclaw
Sierpinski. He was old already then and rather absent-minded. Once he
had to move to a new place for some reason. His wife wife didn't trust
him very much, so when they stood down on the street with all their
things, she said:
- Now, you stand here and watch our ten trunks, while I go and get a
taxi.
She left and left him there, eyes somewhat glazed and humming
absently. Some minutes later she returned, presumably having called
for a taxi. Says Mr. Sierpinski (possibly with a glint in his eye):
- I thought you said there were ten trunks, but I've only counted to nine.
- No, they're TEN!
- No, count them: 0, 1, 2, ..."
-------------------------------------------------------------------------------
Philosopher: "Resolution of the continuum hypothesis will have
profound implications to all of science."
Physicist: "Not quite. Physics is well on its way without those
mythical `foundations'. Just give us serviceable mathematics."
Computer Scientist:
"Who cares? Everything in this Universe seems to be finite
anyway. Besides, I'm too busy debugging my Pascal programs."
Mathematician:
"Forget all that! Just make your formulae as aesthetically
pleasing as possible!"
-------------------------------------------------------------------------------
Definition:
Jogging girl scout = Brownian motion.
-------------------------------------------------------------------------------
lim sin(x)
n --> oo ------ = 6
n
Proof: cancel the n in the numerator and denominator.
-------------------------------------------------------------------------------
Two male 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'.
Her: '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'!
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Last modified 18-November-2001.