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Math is turning bad
"Psst, c'mere," said the shifty-eyed man wearing a long black
trenchcoat, as he beckoned me off the rainy street into a damp dark alley. I
followed.
"What are you selling?" I asked.
"Geometrical algebra drugs."
"Huh!?"
"Geometry drugs. Ya got your uppers, your downers, your sidewaysers, your
inside-outers..."
"Stop right there," I interrupted. "I've never heard of inside-outers."
"Oh, man, you'll love 'em. Makes you feel like M.C. ever-lovin' Escher on a
particularly weird day."
"Go on..."
"OK, your inside-outers, your arbitrary bilinear mappers, and here, heh, here
are the best ones," he said, pulling out a large clear bottle of orange pills.
"What are those, then?" I asked.
"Givens transformers. They'll rotate you about more planes than you even knew
existed."
"Sounds gross. What about those bilinear mappers?"
"There's a whole variety of them. Here's one you'll love -- they call it 'One
Over Z' on the street. Take one of these little bad boys and you'll be on
speaking terms with the Point at Infinity."
Two plus two is five
"First and above all he was a logician. At least thirty-five years of
the half-century or so of his existence had been devoted exclusively to proving
that two and two always equal four, except in unusual cases, where they equal
three or five, as the case may be." -- Jacques Futrelle, "The Problem of Cell
13"
Most mathematicians are familiar with -- or have at least seen references in the
literature to -- the equation 2 + 2 = 4. However, the less well known equation 2
+ 2 = 5 also has a rich, complex history behind it. Like any other complex
quantitiy, this history has a real part and an imaginary part; we shall deal
exclusively with the latter here.
Many cultures, in their early mathematical development, discovered the equation
2 + 2 = 5. For example, consider the Bolb tribe, descended from the Incas of
South America. The Bolbs counted by tying knots in ropes. They quickly realized
that when a 2-knot rope is put together with another 2-knot rope, a 5-knot rope
results.
Recent findings indicate that the Pythagorean Brotherhood discovered a proof
that 2 + 2 = 5, but the proof never got written up. Contrary to what one might
expect, the proof's nonappearance was not caused by a cover-up such as the
Pythagoreans attempted with the irrationality of the square root of two. Rather,
they simply could not pay for the necessary scribe service. They had lost their
grant money due to the protests of an oxen-rights activist who objected to the
Brotherhood's method of celebrating the discovery of theorems. Thus it was that
only the equation 2 + 2 = 4 was used in Euclid's "Elements," and nothing more
was heard of 2 + 2 = 5 for several centuries.
Around A.D. 1200 Leonardo of Pisa (Fibonacci) discovered that a few weeks after
putting 2 male rabbits plus 2 female rabbits in the same cage, he ended up with
considerably more than 4 rabbits. Fearing that too strong a challenge to the
value 4 given in Euclid would meet with opposition, Leonardo conservatively
stated, "2 + 2 is more like 5 than 4." Even this cautious rendition of his data
was roundly condemned and earned Leonardo the nickname "Blockhead." By the way,
his practice of underestimating the number of rabbits persisted; his celebrated
model of rabbit populations had each birth consisting of only two babies, a
gross underestimate if ever there was one.
Some 400 years later, the thread was picked up once more, this time by the
French mathematicians. Descartes announced, "I think 2 + 2 = 5; therefore it
does." However, others objected that his argument was somewhat less than totally
rigorous. Apparently, Fermat had a more rigorous proof which was to appear as
part of a book, but it and other material were cut by the editor so that the
book could be printed with wider margins.
Between the fact that no definitive proof of 2 + 2 = 5 was available and the
excitement of the development of calculus, by 1700 mathematicians had again lost
interest in the equation. In fact, the only known 18th-century reference to 2 +
2 = 5 is due to the philosopher Bishop Berkeley who, upon discovering it in an
old manuscript, wryly commented, "Well, now I know where all the departed
quantities went to -- the right-hand side of this equation." That witticism so
impressed California intellectuals that they named a university town after him.
But in the early to middle 1800's, 2 + 2 began to take on great significance.
Riemann developed an arithmetic in which 2 + 2 = 5, paralleling the Euclidean 2
+ 2 = 4 arithmetic. Moreover, during this period Gauss produced an arithmetic in
which 2 + 2 = 3. Naturally, there ensued decades of great confusion as to the
actual value of 2 + 2. Because of changing opinions on this topic, Kempe's proof
in 1880 of the 4-color theorem was deemed 11 years later to yield, instead, the
5-color theorem. Dedekind entered the debate with an article entitled "Was ist
und was soll 2 + 2?"
Frege thought he had settled the question while preparing a condensed version of
his "Begriffsschrift." This condensation, entitled "Die Kleine Begriffsschrift
(The Short Schrift)," contained what he considered to be a definitive proof of 2
+ 2 = 5. But then Frege received a letter from Bertrand Russell, reminding him
that in "Grundbeefen der Mathematik" Frege had proved that 2 + 2 = 4. This
contradiction so discouraged Frege that he abandoned mathematics altogether and
went into university administration.
Faced with this profound and bewildering foundational question of the value of 2
+ 2, mathematicians followed the reasonable course of action: they just ignored
the whole thing. And so everyone reverted to 2 + 2 = 4 with nothing being done
with its rival equation during the 20th century. There had been rumors that
Bourbaki was planning to devote a volume to 2 + 2 = 5 (the first forty pages
taken up by the symbolic expression for the number five), but those rumor
remained unconfirmed. Recently, though, there have been reported
computer-assisted proofs that 2 + 2 = 5, typically involving computers belonging
to utility companies. Perhaps the 21st century will see yet another revival of
this historic equation.
The above was written by Houston Euler.
Story about infinity
A very large mathematical convention was held in Las Vegas. The
conventioneers filled two hotels, each with an infinite number of rooms. The
hotels were across the street from each other and were owned by brothers. One
evening, while everyone was out at a bar-b-que, one of the hotels burned to the
ground. The brothers got together and worked out a plan. In the remaining hotel,
they moved all guests to twice their room number -- room 101 moved to 202, room
1234 moved to room 2468, etc. Then all the odd number rooms were empty, and
there were an infinite number of odd rooms. So the guests from the other hotel
moved into them
Refrigerate elephants
Analysis:
1. Differentiate it and put into the refrig. Then integrate it in the refrig.
2. Redefine the measure on the referigerator (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.
Topology:
1. Have it swallow the refrig. and turn inside out.
2. Make a refrig. with the Klein bottle.
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 referigerator 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, R^3.
Linear algebra:
1. Put just its basis and span it in the refrig.
2. Show that 1% of the elephant will fit inside the refrigerator. By linearity,
x% will fit for any x.
Affine geometry:
There is an affine transformation putting the elephant into the refrigerator.
Set theory:
1. It's very easy! Refrigerator = { elephant } 2) The elephant and the interior
of the refrigerator both have cardinality c.
Geometry:
Declare the following:
Axiom 1. An elephant can be put into a refrigerator.
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.
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.
Debate about the box
An engineer, a physicist, and a mathematician are trying to set up a
fenced-in area for some sheep, but they have a limited amount of building
material. The engineer gets up first and makes a square fence with the material,
reasoning that it's a pretty good working solution. "No no," says the physicist,
"there's a better way." He takes the fence and makes a circular pen, showing how
it encompasses the maximum possible space with the given material.
Then the mathematician speaks up: "No, no, there's an even better way." To the
others' amusement he proceeds to construct a little tiny fence around himself,
then declares:
"I define myself to be on the outside."
The math one-liners
Math problems? Call 1-800-[(10x)(13i)^2]-[sin(xy)/2.362x].
If parallel lines meet at infinity - infinity must be a very noisy place with
all those lines crashing together!
Maths Teacher: Now suppose the number of sheep is x...
Student: Yes sir, but what happens if the number of sheep is not x?
Zenophobia: the irrational fear of convergent sequences.
Philosophy is a game with objectives and no rules. Mathematics is a game with
rules and no objectives.
If I had only one day left to live, I would live it in my statistics class: it
would seem so much longer.
The birthday study
It is proven that the celebration of birthdays is healthy. Statistics
show that those people who celebrate the most birthdays become the oldest. -- S.
den Hartog, Ph D. Thesis Universtity of Groningen.
The results of statistics
1. Ten percent of all car thieves are left-handed
2. All polar bears are left-handed
3. If your car is stolen, there's a 10 percent chance it was taken by a Polar
bear
1. 39 percent of unemployed men wear spectacles
2. 80 percent of employed men wear spectacles
3. Work stuffs up your eyesight
1. All dogs are animals
2. All cats are animals
3. Therefore, all dogs are cats
1. A total of 4000 cans are opened around the world every second
2. Ten babies are conceived around the world every second
3. Each time you open a can, you stand a 1 in 400 chance of becoming pregnant
Risk of plane bombs
A mathematician and a non-mathematician are sitting in an airport
hall waiting for their flight to go. The non has terrible flight panic.
"Hey, don't worry, it's just every 10000th flight that crashes."
"1:10000? So much? Then it surely will be mine!"
"Well, there is an easy way out. Simply take the next plane. It's much more
probable that you go from a crashing to a non-crashing plane than the other way
round. So you are already at 1:10000 squared."
Statistical one-liners
A new government 10 year survey cost $3,000,000,000 revealed that 3/4
of the people in America make up 75% of the population.
According to recent surveys, 51% of the people are in the majority.
Did you know that 87.166253% of all statistics claim a precision of results that
is not justified by the method employed?
80% of all statistics quoted to prove a point are made up on the spot.
According to a recent survey, 33 of the people say they participate in surveys.
Q: What do you call a statistician on drugs?
A: A high flyer.
Q: How many statisticians does it take to change a lightbulb?
A: 1-3, alpha = .05
There is no truth to the allegation that statisticians are mean. They are just
your standard normal deviates.
Q: Did you hear about the statistician who invented a device to measure the
weight of trees?
A: It's referred to as the log scale.
Q: Did you hear about the statistician who took the Dale Carnegie course?
A: He improved his confidence from .95 to .99.
Q: Why don't statisticians like to model new clothes?
A: Lack of fit.
Q: Did you hear about the statistician who was thrown in jail?
A: He now has zero degrees of freedom.
Statisticians must stay away from children's toys because they regress so
easily.
The only time a pie chart is appropriate is at a baker's convention.
Never show a bar chart at an AA meeting.
Old statisticians never die, they just undergo a transformation.
Q: How do you tell one bathroom full of statisticians from another?
A: Check the p-value.
Q: Did you hear about the statistician who made a career change and became an
surgeon specializing in ob/gyn?
A: His specialty was histerectograms.
The most important statistic for car manufacturers is autocorrelation.
Some statisticians don't drink because they are t-test totalers. Others drink
the hard stuff as evidenced by the proliferation of box-and-whiskey plots.
Underwater ship builders are concerned with sub-optimization.
The Lipton Company is big on statistics--especially t-tests.
Purchasing the shoes
A shoeseller meets a mathematician and complains that he does not
know what size shoes to buy. "No problem," says the mathematician, "there is a
simple equation for that," and he shows him the Gaussian normal distribution.
The shoeseller stares some time at het equation and asks, "What is that symbol?"
"That is the Greek letter pi." "What is pi?" "That is the ratio between the
circumference and the diameter of a circle." Upon this the shoeseller cries out:
"What does a circle have to do with shoes?!"
Answering machine
Hello, this is probably 438-9012, yes, the house of the famous
statistician. I'm probably not at home, or not wanting to answer the phone, most
probably the latter, according to my latest calculations. Supposing that the
universe doesn't end in the next 30 seconds, the odds of which I'm still trying
to calculate, you can leave your name, phone number, and message, and I'll
probably phone you back. So far the probability of that is about 0.645. Have a
nice day.
Worries while flying
Two statisticians were travelling in an airplane from LA to New York.
About an hour into the flight, the pilot announced that they had lost an engine,
but don't worry, there are three left.
However, instead of 5 hours it would take 7 hours to get to New York. A little
later, he announced that a second engine failed, and they still had two left,
but it would take 10 hours to get to New York.
Somewhat later, the pilot again came on the intercom and announced that a third
engine had died. Never fear, he announced, because the plane could fly on a
single engine.
However, it would now take 18 hours to get to new York. At this point, one
statistician turned to the other and said, "Gee, I hope we don't lose that last
engine, or we'll be up here forever!"
Misunderstood people
1. They speak only the Greek language.
2. They usually have long threatening names such as Bonferonni, Tchebycheff,
Schatzoff, Hotelling, and Godambe. Where are the statisticians with names such
as Smith, Brown, or Johnson?
3. They are fond of all snakes and typically own as a pet a large South American
snake called an ANOCOVA.
4. For perverse reasons, rather than view a matrix right side up they prefer to
invert it.
5. Rather than moonlighting by holding Amway parties they earn a few extra bucks
by holding pocket-protector parties.
6. They are frequently seen in their back yards on clear nights gazing through
powerful amateur telescopes looking for distant star constellations called
ANOVA's.
7. They are 99% confident that sleep can not be induced in an introductory
statistics class by lecturing on z-scores.
8. Their idea of a scenic and exotic trip is traveling three standard deviations
above the mean in a normal distribution.
9. They manifest many psychological disorders because as young statisticians
many of their statistical hypotheses were rejected.
10. They express a deap-seated fear that society will someday construct tests
that will enable everyone to make the same score. Without variation or
individual differences the field of statistics has no real function and a
statistician becomes a penniless ward of the state.
Reducing travel risk
There was this statistics student who, when driving his car, would
always accelerate hard before coming to any junction, whizz straight over it ,
then slow down again once he'd got over it. One day, he took a passenger, who
was understandably unnerved by his driving style, and asked him why he went so
fast over junctions. The statistics student replied, "Well, statistically
speaking, you are far more likely to have an accident at a junction, so I just
make sure that I spend less time there."
The fate of marriages
It is often cited that there are half as many divorces as marriages
in the US, so one concludes that average marriages have a 50% chance of ending
by divorce. While I was a graduate student, among my peers there were twice as
many divorces as marriages, leading us to conclude that average marriages would
end twice...