1. Definitions
Let's start with general definitions.
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Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.
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"A mathematician is a device for turning coffee into theorems" (P. Erdos)
Addendum: American coffee is good for lemmas.
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An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care.
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Old mathematicians never die; they just lose some of their functions.
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Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language, and forthwith it means something entirely different. -- Goethe
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What is a rigorous definition of rigor?
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There is no logical foundation of mathematics, and Gödel has proved it!
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I do not think -- therefore I am not.
Here is the illustration of this principle:
One evening Rene Descartes went to relax at a local tavern. The tender approached and said, "Ah, good evening Monsieur Descartes! Shall I serve you the usual drink?". Descartes replied, "I think not.", and promptly vanished.
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A topologist is a person who doesn't know the difference between a coffee cup and a doughnut.
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A mathematician is a blind man in a dark room looking for a black cat which isn't there. (Charles R Darwin)
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A statistician is someone who is good with numbers but lacks the personality to be an accountant.
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Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas.
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A law of conservation of difficulties: there is no easy way to prove a deep result.
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A tragedy of mathematics is a beautiful conjecture ruined by an ugly fact.
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Algebraic symbols are used when you do not know what you are talking about.
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Philosophy is a game with objectives and no rules.
Mathematics is a game with rules and no objectives.
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The actual quote from the Webster dictionary:
trillion n
syn SCAD, gob(s), heap, jillion, load(s), million, oodles, quantities, thousand, wad(s)
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Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato)
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The difference between an introvert and extrovert mathematicians is: An introvert mathematician looks at his shoes while talking to you. An extrovert mathematician looks at your shoes.
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A bit of theology.
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Math is the language God used to write the universe.
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Asked if he believes in one God, a mathematician answered:
" Yes, up to isomorphism."
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Medicine makes people ill, mathematics make them sad and theology makes them sinful. (Martin Luther)
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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)
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He who can properly define and divide is to be considered a god. (Plato)
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"God geometrizes" says Plato.
and here is the analytical continuation of this saying:
Biologists think they are biochemists,
Biochemists think they are Physical Chemists,
Physical Chemists think they are Physicists,
Physicists think they are Gods,
And God thinks he is a Mathematician.
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Physicists defer only to mathematicians, mathematicians defer only to God.
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2. A mathematician and ...
The following sketches show our dedication to abstract thinking in the most unusual situations and strong belief in the universality of mathematical methods. Mathematicians are always impatient and intelligent.
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A mathematician and a Wall street broker went to races. The broker suggested to bet $10,000 on a horse. The mathematician was sceptical, saying that he wanted first to understand the rules, to look on horses, etc. The broker whispered that he knew a secret algorithm for the success, but he could not convince the mathematician.
"You are too theoretical," he said and bet on a horse. Surely, that horse came first bringing him a lot of money. Triumphantly, he exclaimed:
"I told you, I knew the secret!"
"What is your secret?" the mathematician asked.
"It is rather easy. I have two kids, three and five year old. I sum up their ages and I bet on number nine."
"But, three and five is eight," the mathematician protested.
"I told you, you are too theoretical!" the broker replied, "Haven't I just shown experimentally, that my calculation is correct! 3+5=9!"
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A mathematician, a physicist, an engineer went again to the races 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, "first I assumed all the horses were identical and spherical..."
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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 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, etc. 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.
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A physicist and a mathematician are sitting in a faculty lounge. Suddenly, the coffee machine catches on fire. The physicist grabs a bucket and leap towards the sink, filled 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, got a bucket, hands the bucket to the physicist, thus reducing the problem to a previously solved one.
Another version:
A mathematician and an engineer are on desert island. They find two palm trees with one coconut each. The engineer shins up one tree, gets the coconut, eats. The mathematician shins 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."
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A biologist, a physicist and a mathematician were sitting in a street cafe watching the crowd. Across the street they saw a man and a woman entering a building. Ten minutes 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.
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Several scientists 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, then announces: "I don't know what the answer is, but I can tell you, an answer exists!".
Philosopher smiles: "But what do you mean by 2 * 2 ?"
Logician replies: "Please define 2 * 2 more precisely."
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An chemist, a physicist, and a mathematician are stranded on an island when a can of food rools ashore. The chemist and the physicist comes up with many ingenious ways to open the can. Then suddenly the mathematician gets a bright idea: "Assume we have a can opener ..."
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A mathematician is asked to design a table. He first designs a table with no legs. Then he designs a table with infinitely many legs. He spend the rest of his life generalizing the results for the table with N legs (where N is not necessarily a natural number).
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Several scientists were all posed the following question: "What is pi ?"
The engineer said: "It is approximately 3 and 1/7"
The physicist said: "It is 3.14159"
The mathematician thought a bit, and replied "It is equal to pi".
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An engineer, a physicist and a mathematician were asked to hammer a nail into a wall.
The engineer went to build a Universal Automatic Nailer -- a device able to hammer every possible nail into every possible wall.
The physicist conducted series of experiments on strength of hammers, nails, and walls and developed a revolutionary technology of ultra-sonic nail hammering at super-low temperature.
The mathematician generalized the problem to a N dimensional problem of penetration of a knotted one dimensional nail into a N-1 dimensional hyper-wall. Several fundamental theorems are proved. Of course, the problem is too rich to suggest a possibility of a simple solution, even the existence of a solution is far from obvious.
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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!"
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A Mathematician (M) and an Engineer (E) attend a lecture by a Physicist. The topic concerns Kulza-Klein theories involving physical processes that occur in spaces with dimensions of 9, 12 and even higher. The M is sitting, clearly enjoying the lecture, while the E is frowning and looking generally confused and puzzled. By the end the E has a terrible headache. At the end, the M comments about the wonderful lecture.
E: "How do you understand this stuff?"
M: "I just visualize the process"
E: "How can you POSSIBLY visualize something that occurs in 9-dimensional space?"
M: "Easy, first visualize it in N-dimensional space, then let N go to 9"
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A team of engineers were required to measure the height of a flag pole. They only had a measuring tape, and were getting quite frustrated trying to keep the tape along the pole. It kept falling down, etc. A mathematician comes along, finds out their problem, and proceeds to remove the pole from the ground and measure it easily. When he leaves, one engineer says to the other: "Just like a mathematician! We need to know the height, and he gives us the length!"
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A mathematician and a physicist agree to a psychological experiment. The (hungry) mathematician is put in a chair in a large empty room and his favorite meal, perfectly prepared, is placed at the other end of the room. The psychologist explains, "You are to remain in your chair. Every minute, I will move your chair to a position halfway between its current location and the meal." The mathematician looks at the psychologist in disgust. "What? I'm not going to go through this. You know I'll never reach the food!" And he gets up and storms out.
The psychologist ushers the physicist in. He explains the situation, and the physicist's eyes light up and he starts drooling. The psychologist is a bit confused. "Don't you realize that you'll never reach the food?" T he physicist smiles and replies: "Of course! But I'll get close enough for all practical purposes!"
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One day a farmer called up an engineer, a physicist, and a mathematician and asked them to fence of 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."
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The physicist and the engineer are in a hot-air balloon. Soon, they find themselves lost in a canyon somewhere. They yell out for help: "Helllloooooo! Where are we?"
15 minutes later, they hear an echoing voice: "Helllloooooo! You're in a hot-air balloon!!"
The physicist says, "That must have been a mathematician."
The engineer asks, "Why do you say that?"
The physicist replied: "The answer was absolutely correct, and it was utterly useless."
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Several scientists were asked to prove that all odd integers higher than 2 are prime.
Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, and by induction - every odd integer higher than 2 is a prime.
Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime. Just to be sure, try several randomly chosen numbers: 17 is a prime, 23 is a prime...
Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an approximation to a prime, 11 is a prime,...
Programmer (reading the output on the screen): 3 is a prime, 3 is a prime, 3 a is prime, 3 is a prime....
Biologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 -- results have not arrived yet,...
Psychologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime but tries to suppress it,...
Chemist (or Dan Quayle): What's a prime?
Politician: "Some numbers are prime.. but the goal is to create a kinder, gentler society where all numbers are prime... "
Programmer: "Wait a minute, I think I have an algorithm from Knuth on finding prime numbers... just a little bit longer, I've found the last bug... no, that's not it... ya know, I think there may be a compiler bug here - oh, did you want IEEE-998.0334 rounding or not? - was that in the spec? - hold on, I've almost got it - I was up all night working on this program, ya know... now if management would just get me that new workstation tha just came out, I'd be done by now... etc., etc. ..."
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Dean, to the physics department. "Why do I always have to give you guys so much money, for laboratories and expensive equipment and stuff. Why couldn't you be like the math. department - all they need is money for pencils, paper and waste-paper baskets. Or even better, like the philosophy department. All they need are pencils and paper."
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A mathematician, an engineer, and a chemist were walking down the road when they saw a pile of cans of beer. Unfortunately, they were the old-fashioned cans that do not have the tab at the top. One of them proposed that they split up and find can openers. The chemist went to his lab and concocted a magical chemical that dissolves the can top in an instant and evaporates the next instant so that the beer inside is not affected. The engineer went to his workshop and created a new HyperOpener that can open 25 cans per second.
They went back to the pile with their inventions and found the mathematician finishing the last can of beer. "How did you manage that?" they asked in astonishment. The mathematician answered, "Oh, well, I assumed they were open and went from there."
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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.
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What is the difference between a Psychotic, a Neurotic and a mathematician? A Psychotic believes that 2+2=5. A Neurotic knows that 2+2=4, but it kills him. A mathematician simply changes the base.
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To mathematicians, solutions mean finding the answers. But to chemists, solutions are things that are still all mixed up.
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3. Math. education
These sketches demonstrate how desperately we want to push the math into the public education, and the struggle and passion of math. students.
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The Evolution of Math Teaching
1960s: A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price. What is his profit?
1970s: A farmer sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price, that is, $8. What is his profit?
1970s (new math): A farmer exchanges a set P of potatoes with set M of money. The cardinality of the set M is equal to 10, and each element of M is worth $1. Draw ten big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and give the answer to the question: What is the cardinality of the set of profits?
1980s: A farmer sells a bag of potatoes for $10. His production costs are $8, and his profit is $2. Underline the word "potatoes" and discuss with your classmates.
1990s: A farmer sells a bag of potatoes for $10. His or her production costs are 0.80 of his or her revenue. On your calculator, graph revenue vs. costs. Run the POTATO program to determine the profit. Discuss the result with students in your group. Write a brief essay that analyzes this example in the real world of economics.
(Anon: adapted from The American Mathematical Monthly, Vol. 101, No. 5, May 1994 (Reprinted by STan Kelly-Bootle in Unix Review, Oct 94)
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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 four-dimensional 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.
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.
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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!"
Frankly, this story is sad. Indeed, the blonde waitress solves the problem correctly, better than the poser. But he is a mathematician, and she cleans tables. Funny, isn't it?
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A mathematician, native Texan, once was asked in his class: "What is mathematics good for?" He replied: "This question makes me sick. Like when you show somebody the Grand Canyon for the first time, and he asks you `What's is good for?' What would you do? Why, you would kick the guy off the cliff".
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A somewhat advanced society has figured how to package basic knowledge in pill form.
A student, needing some learning, goes to the pharmacy and asks what kind of knowledge pills are available. The pharmacist says "Here's a pill for English literature." The student takes the pill and swallows it and has new knowledge about English literature!
"What else do you have?" asks the student.
"Well, I have pills for art history, biology, and world history," replies the pharmacist.
The student asks for these, and swallows them and has new knowledge about those subjects.
Then the student asks, "Do you have a pill for math?"
The pharmacist says "Wait just a moment", and goes back into the storeroom and brings back a whopper of a pill and plunks it on the counter.
"I have to take that huge pill for math?" inquires the student.
The pharmacist replied "Well, you know math always was a little hard to swallow."
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Golden rule for math teachers: You must tell the truth, and nothing but the truth, but not the whole truth.
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A math professor is one who talks in someone else's sleep.
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Q:What do you get when you add 2 apples to 3 apples?
A:Answer: A senior high school math problem.
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Teacher: Now suppose the number of sheep is x...
Student: Yes sir, but what happens if the number of sheep is not x?
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Mathematician U. was a great friend of his five-year old grandson. They discused everything including math and U. was very proud of the boys math talents. The child went to kindergarden; In two weeks the he ask U.to help with the difficult math problem: "There are four airplanes flying, then two more airplanes join them. How many airplanes are flying now? U. was very disappointed by the simplicity of the problem. "What confuses you?" he asked. The child says: " I know, of course, that 4 + 2 =6, but I cannot figure out what the airplanes have do with this!"
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Quotes from math students and lecturers
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"This is a one line proof...if we start sufficiently far to the left."
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"The problems for the exam will be similar to the discussed in the class. Of course, the numbers will be different. But not all of them. Pi will still be 3.14159... "
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"Roses are red,
Violets are blue,
Greens' functions are boring
And so are Fourier transforms."
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"Sex and drugs? They're nothing compared with a good proof!"
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4. Seminar semantics, etc.
Next passages contain little professional secrets. They reflect the conflict between the dreams of classical clear presentations, the complexity of modern math problems, and the survival tactics of the authors.
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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 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 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.
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Dictionary of Definitions of Terms Commonly Used in Math. lectures.
The following is a guide to terms which are commonly used but rarely defined. In the search for proper definitions for these terms we found no authoritative, nor even recognized, source. Thus, we followed the advice of mathematicians handed down from time immortal: "Wing 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.
OBVIOUSLY:
I hope you weren't sleeping when we discussed this earlier, because I refuse to repeat it.
RECALL:
I shouldn't have to tell you this, but for those of you who erase your memory tapes after every test...
WLOG (Without Loss Of Generality):
I'm not about to do all the possible cases, so I'll do one and let you figure out the rest.
IT CAN EASILY BE SHOWN:
Even you, in your finite wisdom, should be able to prove this without me holding your hand.
CHECK or CHECK FOR YOURSELF:
This is the boring part of the proof, so you can do it on your own time.
SKETCH OF A PROOF:
I couldn't verify all the details, so I'll break it down into the parts I couldn't prove.
HINT:
The hardest of several possible ways to do a proof.
BRUTE FORCE (AND IGNORANCE):
Four special cases, three counting arguments, two long inductions, "and a partridge in a pair tree."
SOFT PROOF:
One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.
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.
CANONICAL FORM:
4 out of 5 mathematicians surveyed recommended this as the final form for their students who choose to finish.
TFAE (The Following Are Equivalent):
If I say this it means that, and if I say that it means the other thing, and if I say the other thing...
BY A PREVIOUS THEOREM:
I don't remember how it goes (come to think of it I'm not really sure we did this at all), but if I stated it right (or at all), then the rest of this follows.
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.
LET'S TALK THROUGH IT:
I don't want to write it on the board lest I make a mistake.
PROCEED FORMALLY:
Manipulate symbols by the rules without any hint of their true meaning (popular in pure math courses).
QUANTIFY:
I can't find anything wrong with your proof except that it won't work if x is a moon of Jupiter (Popular in applied math courses).
PROOF OMITTED:
Trust me, It's true.
Traditional - contemporary math dictionary.
WHAT'S OUT AND WHAT'S IN FOR MATHEMATICAL TERMS
by
Michael Stueben (November 7, 1994)
Today it is considered an egregious faux pas to speak or write in the
crude antedated terms of our grandfathers. To assist the isolated
student and the less sophisticated teacher, I have prepared the
following list of currently fashionable mathematical terms in
academia. I pass this list on to the general public as a matter of
charity and in the hope that it will lead to more refined elucidation
from young scholars.
thinking: hypothesizing.
proof by contradiction or indirect proof: reductio ad absurdum.
mistake: non sequitur.
starting place: handle.
with corresponding changes: mutatis mutandis.
counterexample: pathological exception.
consequently: ipso facto.
swallowing results: digesting proofs.
therefore: ergo.
has an easy-to-understand, but hard-to-find solution: obvious.
has two easy-to-understand, but hard-to-find solutions: trivial.
truth: tautology.
empty: vacuous.
drill problems: plug-and-chug work.
criteria: rubric.
example: substantive
instantiation.
similar structure: homomorphic.
very similar structure: isomorphic.
same area: isometric.
arithmetic: number theory.
count: enumerate.
one: unity.
generally/specifically: globally/locally.
constant: invariant.
bonus result: corollary.
distance: metric measure.
several: a plurality.
function/argument: operator/operand.
separation/joining: bifurcation/confluence.
fourth power or quartic: biquadratic.
random: stochastic.
unique condition: a singularity.
uniqueness: unicity.
tends to zero: vanishes.
tip-top point: apex.
half-closed: half-open.
concave: non-convex.
rectangular prisms: parallelepipeds.
perpendicular (adj.): orthogonal.
perpendicular (n.): normal.
Euclid: Descartes.
Fermat: Wiles.
path: trajectory.
shift: rectilinear translation.
similar: homologous.
very similar: congruent.
whopper-jawed: skew or oblique.
change direction: perturb.
join: concatenate.
approximate to two or more places: accurate.
high school geometry or plane geometry: geometry of the
Euclidean plane
under the Pythagorean metric.
clever scheme: algorithm.
initialize to zero: zeroize.
* : splat.
{ : squiggle.
decimal: denary.
alphabetical order: lexical order.
a divide-and-conquer method: an algorithm of
logarithmic
order.
student ID numbers: witty passwords.
numerology and number sophistry: descriptive statistics
Special thanks to Peter Braxton who got me started
writing this stuff and who contributed five of
the items above.
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Professional secrets.
The highest moments in the life of a mathematician are the first few moments after one has proved the result, but before one finds the mistake.
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Golden rule of deriving: never trust any result that was proved after 11 pm.
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Relations between pure and applied mathematicians are based on trust and understanding. Namely, pure mathematicians do not trust applied mathematicians, and applied mathematicians do not understand pure mathematicians.
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How dare of you to think that I am an analyst!
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Some mathematicians become so tense these days that they that they do not go to sleep during seminars.
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If I have seen farther than others, it is because I was standing on the shoulders of giants.
-- Isaac Newton
In the sciences, we are now uniquely priviledged to sit side by side with the giants on whose shoulders we stand.
-- Gerald Holton
If I have not seen as far as others, it is because giants were standing on my shoulders.
-- Hal Abelson
Mathematicians stand on each other's shoulders.
-- Gauss
Mathemeticians stand on each other's shoulders while computer scientists stand on each other's toes.
-- Richard Hamming
It has been said that physicists stand on one another's shoulders. If this is the case, then programmers stand on one another's toes, and software engineers dig each other's graves.
-- Unknown
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These days, even the most pure and abstract mathematics is in danger to be applied.
To the top
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5. Theorems
Here, the powerful mathematical methods are successively applied to the "real life problems".
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Interesting 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.
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Discovery.
Mathematicians have announced the existence of a new whole number which lies between 27 and 28. "We don't know why it's there or what it does," says Cambridge mathematician, Dr. Hilliard Haliard, "we only know that it doesn't behave properly when put into equations, and that it is divisible by six, though only once."
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Theorem:
There are two groups of people in the world; those who believe that the world can be divided into two groups of people, and those who don't.
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Theorem:
The world is divided into two classes:
people who say "The world is divided into two classes",
and people who say: The world is divided into two classes:
people who say: "The world is divided into two classes",
and people who say: The world is divided into two classes:
people who say ...
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There are three kinds of people in the world; those who can count and those who can't.
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There are 10 kinds of people in the world, those who understand binary math, and those who don't.
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"The world is everywhere dense with idiots."
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Cat 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.
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Salary Theorem
The less you know, the more you make.
Proof:
Postulate 1: Knowledge is Power.
Postulate 2: Time is Money.
As every engineer knows: Power = Work / Time
And since Knowledge = Power and Time = Money
It is therefore true that Knowledge = Work / Money .
Solving for Money, we get:
Money = Work / Knowledge
Thus, as Knowledge approaches zero, Money approaches infinity, regardless of the amount of Work done.
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The cherry theorem (a puzzle that reminds some of calculus theorems)
Q: What is a small, red, round thing that has a cherry pit inside?
A: A cherry.
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Notes on the horse colors problem
Lemma 1. All horses are the same color. (Proof by induction)
Proof. It is obvious that one horse is the same color. Let us assume the proposition P(k) that k horses are the same color and use this to imply that k+1 horses are the same color. Given the set of k+1 horses, we remove one horse; then the remaining k horses are the same color, by hypothesis. We remove another horse and replace the first; the k horses, by hypothesis, are again the same color. We repeat this until by exhaustion the k+1 sets of k horses have been shown to be the same color. It follows that since every horse is the same color 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 color.
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 certainly 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 number 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 color, and by the lemma that does not exist.
Corollary 1. Everything is the same color.
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 universally-quantified conditional 'For all x, if x is a horse, then x is the same color,' 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 color.'
Corollary 2. Everything is white.
Proof. If a sentential formula in x is logically true, then any particular substitution instance of it is a true sentence. In particular then: 'for all x, if x is an elephant, then x is the same color' is true. Now it is manifestly axiomatic that white elephants exist (for proof by blatant assertion consult Mark Twain 'The Stolen White Elephant'). 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 consequent 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.
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The mathematical theory of big game hunting (Aug-Sept. AMM, 446-447, 1938):
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6. Playground
Let's play with math objects!
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An insane mathematician gets on a bus and starts threatening everybody: "I'll integrate you! I'll differentiate you!!!" Everybody gets scared and runs away. Only one lady stays. The guy comes up to her and says: "Aren't you scared, I'll integrate you, I'll differentiate you!!!" The lady calmly answers: "No, I am not scared, I am e^x ."
More anwanced and more New York style story:
A constant function and e^x are walking on Broadway. Then suddenly the constant function sees a differential operator approaching and runs away. So e^x follows him and asks why the hurry. "Well, you see, there's this differential operator coming this way, and when we meet, he'll differentiate me and nothing will be left of me...!" "Ah," says e^x, "he won't bother ME, I'm e to the x!" and he walks on. Of course he meets the differential operator after a short distance.
e^x: "Hi, I'm e^x"
diff.op.: "Hi, I'm d/dy"
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"The number you have dialed is imaginary. Please rotate your phone 90 degrees and try again."
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The shortest math joke: let epsilon be < 0
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Funny formulas
The limit as 3 goes to 4 of 3^2 is 16.
(For native LaTex speakers: $$\lim_{3 \rightarrow 4} 3^2 = 16$$)
1 + 1 =3, for sufficiently large one's.
The combination of the Einstein and Pythagoras discoveries:
E= m c^2= m ( a^2 + b^2)
2 and 2 is 22
The limit as n goes to infinity of sin (x) /n is 6.
Proof: cancel the n in the numerator and denominator.
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Examples of inverse problems:
Q: To what question is the answer "9W."
A: "Dr. Wiener, do you spell your name with a V?"
Q: To what question is the answer "Dr. Livingstone, I presume."
A: "What is your full name, Dr. Presume?"
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Q: how many times can you subtract 7 from 83, and what is left afterwards?
A: I can subtract it as many times as I want, and it leaves 76 every time.
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A Neanderthal child rode to school with a boy from Hamilton. When his mother found out she said, "What did I tell you? If you commute with a Hamiltonian you'll never evolve!"
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Pope has settled the continuum hypothesis!
He has declared that cardinals above 80 have no powers.
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In modern mathematics, algebra has become so important that numbers will soon only have symbolic meaning.
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A circle is a round straight line with a hole in the middle.
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In the topologic hell the beer is packed in Klein's bottles.
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He thinks he's really smooth, but he's only C^1.
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Q: What is the world's longest song?
A: "Aleph-nought Bottles of Beer on the Wall."
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Q: Why do Computer Scientists get Halloween and Christmas mixed up?
A: Because Oct. 31 = Dec. 25.
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Q: Why did the chicken cross the Moebius strip?
A: To get to the other ... er, um ...
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Q: Why did the mathematician name his dog "Cauchy"?
A: Because he left a residue at every pole.
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Q: What do you get when you cross an elephant and a banana?
A: elephant * banana * sin(theta)
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Q: What do you get if you cross a mosquito with a mountain climber.
A: You can't cross a vector with a scalar.
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Q: What is a compact city?
A: It's a city that can be guarded by finitely many near-sighted policemen.
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A statistician can have his head in an oven and his feet in ice, and he will say that on the average he feels fine.
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Q: Did you hear the one about the statistician?
A: Probably....
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The light bulb problem
Q: How many mathematicians does it take to screw in a light bulb?
A1: None. It's left to the reader as an exercise.
A2: None. A mathematician can't screw in a light bulb, but he can easily prove the work can be done.
A3: One. He gives it to four programmers, thereby reducing the problem to the already solved (ask a programmer, how)
A4: The answer is intuitively obvious
A5: Just one, once you've managed to present the problem in terms he/she is familiar with.
A6: In earlier work, Wiener [1] has shown that one mathematician can change a light bulb.
If k mathematicians can change a light bulb, and if one more simply watches them do it, then k+1 mathematicians will have changed the light bulb.
Therefore, by induction, for all n in the positive integers, n mathematicians can change a light bulb.
Bibliography:
[1] Weiner, Matthew P,...
How many mathematical logicians does it take to replace a lightbulb??
None: They can't do it, but they can prove that it can be done.
How many numerical analysts does it take to replace a lightbulb??
3.9967: (after six iterations).
How many classical geometers does it take to replace a lightbulb??
None: You can't do it with a straight edge and a compass.
How many constructivist mathematicians does it take to replace a lightbulb??
None: They do not believe in infinitesimal rotations.
How many simulationists does it take to replace a lightbulb??
Infinity: Each one builds a fully validated model, but the light actually never goes on.
How many topologists does it take to screw in a lightbulb??
Just one. But what will you do with the doughnut?
How many analysts does it take to screw in a lightbulb??
Three: One to prove existence, one to prove uniqueness and one to derive a nonconstructive algorithm to do it.
How many Bourbakists does it take to replace a lightbulb: ?
Changing a lightbulb is a special case of a more general theorem concerning the maintain 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. Iff 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.
How many professors does it take to replace a lightbulb??
One: With eight research students, two programmers, three post-docs and a secretary to help him.
How many university lecturers does it take to replace a lightbulb??
Four: One to do it and three to co-author the paper.
How many graduate students does it take to replace a lightbulb??
Only one: But it takes nine years.
How many math department administrators does it take to replace a lightbulb?
None: What was wrong with the old one then???
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How we do it ...
Aerodynamicists do it in drag.
Algebraists do it by symbolic manipulation.
Algebraists do it in a ring, in fields, in groups.
Analysts do it continuously and smoothly.
Applied mathematicians do it by computer simulation.
Banach spacers do it completely.
Bayesians do it with improper priors.
Catastrophe theorists do it falling off part of a sheet.
Combinatorists do it as many ways as they can.
Complex analysts do it between the sheets
Computer scientists do it depth-first.
Cosmologists do it in the first three minutes.
Decision theorists do it optimally.
Functional analysts do it with compact support.
Galois theorists do it in a field.
Game theorists do it by dominance or saddle points.
Geometers do it with involutions.
Geometres do it symmetrically.
Graph theorists do it in four colours.
Hilbert spacers do it orthogonally.
Large cardinals do it inaccessibly.
Linear programmers do it with nearest neighbors.
Logicians do it by choice, consistently and completely.
Logicians do it incompletely or inconsistently.
(Logicians do it) or [not (logicians do it)].
Number theorists do it perfectly and rationally.
Mathematical physicists understand the theory of how to do it, but have difficulty obtaining practical results.
Pure mathematicians do it rigorously.
Quantum physicists can either know how fast they do it, or where they do it, but not both.
Real analysts do it almost everywhere
Ring theorists do it non-commutatively.
Set theorists do it with cardinals.
Statisticians probably do it.
Topologists do it openly, in multiply connected domains
Variationists do it locally and globally.
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Cantor did it diagonally.
Fermat tried to do it in the margin, but couldn't fit it in.
Galois did it the night before.
Mðbius 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.
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A SLICE OF PI
******************
3.14159265358979
1640628620899
23172535940
881097566
5432664
09171
036
5
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7. Puns
These jokes are sometimes stupid, but still funny.
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Math and Alcohol don't mix, so... PLEASE DON'T DRINK AND DERIVE
Motto of the society: Mathematicians Against Drunk Deriving
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Q: What's the 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!
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Q: What is the area of a circle?
A: pi R^2?
R: Pie are not square. Pie are round. Cornbread are square.
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Q:What is a proof?
A: One-half percent of alcohol.
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Q:What is a dilemma?
A: A lemma that proves two results.
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Q: What's nonorientable and lives in the sea?
A: Mobius Dick.
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Q: What's yellow and equivalent to the Axiom of Choice.
A: Zorn's Lemon.
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Q: What's purple and commutes?
A: An abelian grape.
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Q: What's yellow, linear, normed and complete?
A: A Bananach space.
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Q: What's a polar bear?
A: A rectangular bear after a coordinate transform.
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Some say the pope is the greatest cardinal.
But others insist this cannot be so, as every pope has a successor.
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Q: How many light bulbs does it take to change a light bulb?
A: One, if it knows its own Goedel number.
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Q: What does the little mermaid wear?
A: An Algebra
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Noah's Ark lands after The Flood and Noah releases all the animals, saying, "Go forth and multiply." Several months pass and Noah decides to check up on the animals. All are doing fine except a pair of snakes. "What's the problem?" asks Noah. "Cut down some trees and let us live there," say the snakes. Noah follows their advice. Several more weeks pass and Noah checks up on the snakes again. He sees lots of little snakes; everybody is happy. Noah says, "So tell me how the trees helped." "Certainly," reply the snakes. "We're adders, and we need logs to multiply."
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Q: Why didn't Newton discover group theory?
A: Because he wasn't Abel.
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In Alaska, where it gets very cold, pi is only 3.00. As you know, everything shrinks in the cold. They call it Eskimo pi.
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How do you prove in three steps that a sheet of paper is a lazy dog?
1. A sheet of paper is an ink-lined plane.
2. An inclined plane is a slope up.
3. A slow pup is a lazy dog.
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A geometer went to the beach to catch the rays and became a TanGent.
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Life is complex. It has real and imaginary components.
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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 the poles off the right side of the plane.
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8. Anecdotes
Next several stories are attributed to real mathematicians. For most of them, it was impossible to check the truthfulness of the story. Therefore the names are often removed.
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In 1915, Emma Noether arrived in Göttingen but was denied the private-docent status. The argument was that a woman cannot attend the University senate (the faculty meetings). Hilbert's reaction was: "Gentlemen! There is nothing wrong to have a woman in the senate. Senate is not a bath."
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After Laplace completed his masterpiece Mecanique Celeste (Mechanics of the Heavens) he presented a copy to his friend Napoleon. Napoleon, who also was a mathematician, after going through the book called in Laplace and said to him: "You have written a book about Mechanics of the Heavens without mentioning God?" Laplace replied: "Sire, I had no need of that hypothesis".
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The following problem can be solved either the easy way or the hard way.
Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?
The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles. That's all there is to it.
When this problem was posed to John von Neumann, he immediately replied, "150 miles."
"It is very strange," said the poser, "but nearly everyone tries to sum the infinite series."
"What do you mean, strange?" asked Von Neumann. "That's how I did it!"
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Another von Neumann quote : Young man, in mathematics you don't understand things, you just get used to them.
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The mathematician S. had to move to a new place. His wife didn't trust him very much, so when they stood down on the street with all their things, she asked him to watch their ten trunks, while she get a taxi. Some minutes later she returned. Said the husband:
"I thought you said there were ten trunks, but I've only counted to nine."
The wife said: "No, they're TEN!"
"But I have counted them: 0, 1, 2, ..."
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N. 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. N. looked blank for a moment, thought, and then answered, "Yes".
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In his lecture, ** formulated a theorem and said: "The proof is obvious". Then he thought for a minute, left the lecture room, returned after 15 minutes and happily concluded: "Indeed, it is obvious!"
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A famous mathematician was to give a keynote speech at a conference. Asked for an advance summary, he said he would present a proof of Fermat's Last Theorem -- but they should keep it under their hats. When he arrived, though, he spoke on a much more prosaic topic. Afterwards the conference organizers asked why he said he'd talk about the theorem and then didn't. He replied this was his standard practice, just in case he was killed on the way to the conference.
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A mathematician about his late colleague:
" He made a lot of mistakes, but he made them in a good direction. I tried to copy this, but I found out that it is very difficult to make good mistakes. "
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This story is attributed to Professor Lev Loytiansky, the stage is in Soviet Union in thirties or forties.
L. organized the seminar in hydrodynamics in his University. Among the regular attendees there were two men in the uniform, obviously military engineers. They never discussed the problems they were working on. But one day they ask L. to help with a math. problem. They explained that the solution of a certain equation oscillated and asked how they should change the coefficients to make it monotonic. L. looked on the equation and said: "Make the wings longer!"
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Students asked ** to exclude a part of the course from the final exam. ** agreed. Encouraged by the easy success, the students asked to skip another part of the course, and ** agreed again, and then again. However, in the end of the term he did include all this material in the exam. The class loudly complained:
"Dr **, you promised us to skip this stuff!"
** answered: "Yes, I did. But I lied!"
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This anecdote is attributed to Landau (the Russian physicist Lev not the Göttingen mathematician Edmund).
Landau's group was discussing a bright new theory, and one of junior colleagues of Landau bragged that he had independently discovered the theory a couple of years ago, but did not bother to publish his finding.
"I would not repeat this claim if I were you," Landau replied: "There is nothing wrong if one has not found
a solution to a particular problem. However, if one has found it but does not publish it, he shows a poor judgment and inability to understand what important is in modern physics".
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9. Limericks
Limericks are always limericks. (The obscene math limericks were mercilessly excluded)
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A mathematician confided
That the M"obius band is one-sided
And you'll get quite a laugh
If you cut one in half
'Cause it stays in one piece when divided.
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A mathematician named Klein
Thought the M"obius band was divine
Said he: If you glue
The edges of two
You'll get a weurd bottles like mine.
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There was a young fellow named Fisk,
A swordsman, exceedingly brisk.
So fast was his action,
The Lorentz contraction
Reduced his rapier to a disk.
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'Tis a favorite project of mine
A new value of pi to assign.
I would fix it at 3
For it's simpler, you see,
Than 3 point 1 4 1 5 9
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If inside a circle a line
Hits the center and goes spine to spine
And the line's length is "d"
the circumference will be
d times 3.14159
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Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed?
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A challenge for many long ages
Had baffled the savants and sages.
Yet at last came the light:
Seems old Fermat was right--
To the margin add 200 pages.
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If (1+x) (real close to 1)
Is raised to the power of 1
Over x, you will find
Here's the value defined:
2.718281...
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Integral z-squared dz
from 1 to the cube root of 3
times the cosine
of three pi over 9
equals log of the cube root of 'e'.
And it's correct, too.
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A burleycque dancer, a pip
Named Virginia, could peel in a zip;
But she read science fiction
and died of constriction
Attempting a Moebius strip.
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This poem was written by John Saxon (an author of math textbooks).
((12 + 144 + 20 + (3 * 4^(1/2))) / 7) + (5 * 11) = 9^2 + 0
A Dozen, a Gross and a Score,
plus three times the square root of four,
divided by seven,
plus five times eleven,
equals nine squared and not a bit more.
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In arctic and tropical climes,
the integers, addition, and times,
taken (mod p) will yield
a full finite field,
as p ranges over the primes.
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A graduate student from Trinity
Computed the cube of infinity;
But it gave him the fidgets
To write down all those digits,
So he dropped math and took up divinity.
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Chebychev said it and I'll say it again:
There's always a prime between n and 2n!
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A conjecture both deep and profound
Is whether the circle is round;
In a paper by Erdo"s,
written in Kurdish,
A counterexample is found.
(Note: Erdo"s is pronounced "Air - dish")
Thanks to Andrej and Elena Cherkaev for this
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