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A mathematician wandered home at 3 AM. His wife became very upset, telling him, "You're late! You said you'd be home by 11:45!" The mathematician replied, "I'm right on time. I said I'd be home by a quarter of twelve."
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Two mathematicians are in a bar. The first one says to the second that the average person knows very little about basic mathematics. The second one disagrees, and claims that most people can cope with a reasonable amount of math.
The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer "one third x cubed."
She repeats "one thir -- dex cue"?
He repeats "one third x cubed".
She asks, "one thir dex cuebd?"
"Yes, that's right," he says.
So she agrees, and goes off mumbling to herself, "one thir dex cuebd...".
The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks "what is the integral of x squared?".
The waitress says "one third x cubed" and while walking away, turns back and says over her shoulder "plus a constant!"
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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.
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Mathematics is made of 50 percent formulas, 50 percent proofs and 50 percent imagination.
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Why did the chicken cross the Möbius strip?
To get to the same side.
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There was once a very smart horse. Anything that was shown it, it mastered easily, until one day, its teachers tried to teach it about rectangular coordinates and it couldn't understand them. All the horse's acquaintances and friends tried to figure out what was the matter and couldn't. Then a new guy looked at the problem and said,
"Of course he can't do it. Why, you're putting Descartes before the horse!"
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HOW TO PROVE IT
proof by example:
The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.
proof by intimidation:
"Trivial."
proof by vigorous handwaving:
Works well in a classroom or seminar setting.
proof by cumbersome notation:
Best done with access to at least four alphabets and special symbols.
proof by exhaustion:
An issue or two of a journal devoted to your proof is useful.
proof by omission:
"The reader may easily supply the details"
"The other 253 cases are analogous"
"..."
proof by obfuscation:
A long 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 funding:
How could three different government agencies be wrong?
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 picture:
A more convincing form of proof by example. Combines well with proof by omission.
proof by vehement assertion:
It is useful to have some kind of authority relation to the audience.
proof by ghost reference:
Nothing even remotely resembling the cited theorem appears in the reference given.
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 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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A quiet little man was brought before a judge. The judge looked down at the man and then at the charges and then down at the little man in amazement. "Can you tell me in your own words what happened?" he asked the man.
"I'm a mathematical logician dealing in the nature of proof."
"Yes, go on," said the astounded judge.
"Well, I was at the library and I found the books I wanted and went to take them out. They told me my library card had expired and I had to get a new one. So I went to the registration office and got in another line. And filled out my forms for another card. And got back in line for my card."
"And?" said the judge.
"And he asked 'Can you prove you are from New York City?' ...So I stabbed him."
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There was a logician who saw a sign on his way to fish that read, "All the worms you want for $1.00." He stopped his car and ordered $2.00 worth.
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The book Dynamic Programming by Richard Bellman is an important, pioneering work in which a group of problems is collected together at the end of some chapters under the heading "Exercises and Research Problems," with extremely trivial questions appearing in the midst of deep, unsolved problems. It is rumored that someone once asked Dr. Bellman how to tell the exercises apart from the research problems, and he replied: "If you can solve it, it is an exercise; otherwise it's a research problem."
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A mathematician is a machine for turning coffee into theorems.
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A conjecture both deep and profound
Is whether a circle is round.
In a paper of Erdös
Written in Kurdish
A counterexample is found.
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Approximately ten excuses for not doing homework:
I accidentally divided by zero and my paper burst into flames.
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 am the square root of negative one.
I took time out to snack on a doughnut and a cup of coffee, and then 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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After Receiving an Invitation to a Mathematicians' Ball:
Augustin Louis Cauchy said he surely will managed to integrate well with everyone.
David Hilbert was afraid he will be pretty spaced out for most of the party.
Paul Erdös asked: "Are epsilons invited too?"
John Forbes Nash insisted on playing n-person zero sum games.
Zeno of Elea said he will come with two friends - Achilles and the tortoise.
Bertrand Russell was wondering: "If the cook only cooks for the guests, who cooks for the cook?"
Kurt Gödel insisted that the invitation is incomplete and never will be.
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Pick-Up Lines to use on Mathematics Chicks
You fascinate me more than the Fundamental Theorem of Calculus.
Are you a differentiable function? Because I'd like to be tangent to your curves!
You and I would add up better than a Riemann sum.
My love for you is a monotonic increasing function of time.
Wanna come back to my room and see my copy of Euclid's "Elements"?
I am equivalent to the Empty Set when you are not with me.
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The Dictionary: what mathematics professors say and what they mean by 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.
It can easily be shown: No more than four hours are needed to prove it.
Check for yourself: This is the boring part of the proof, so you can do it on your own time.
Hint: The hardest of several possible ways to do a proof.
Brute force: Four special cases, three counting arguments and two long inductions.
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.
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.
Proceed formally: Manipulate symbols by the rules without any hint of their true meaning.
Proof omitted: Trust me, It's true.
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Mathematics Revisited
Life is complex. It has real and imaginary components.
What keeps a square from moving? Square roots, of course.
The law of the excluded middle either rules or does not rule.
In the topological hell the beer is packed in Klein's bottles.
To a mathematician, real life is a special case.
I heard that parallel lines actually do meet, but they are very discrete.
In modern mathematics, algebra has become so important that numbers will soon only have symbolic meaning.
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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How mathematicians do it...
Combinatorists do it as many ways as they can.
Combinatorists do it discretely.
(Logicians do it) or [not (logicians do it)].
Logicians do it by symbolic manipulation.
Algebraists do it in groups.
Algebraists do it in a ring.
Algebraists do it in a field.
Analysts do it continuously.
Real analysts do it almost everywhere.
Pure mathematicians do it rigorously.
Topologists do it openly.
Topologists do it on rubber sheets.
Dynamicists do it chaotically.
Mathematicians do it forever if they can do one and can do one more.
Cantor did it diagonally.
Fermat tried to do it in the margin, but couldn't fit it in.
Galois did it the night before.
Möbius 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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You Might Be a Mathematician if...
you are fascinated by the equation .
you know by heart the first fifty digits of .
you have tried to prove Fermat's Last Theorem.
you know ten ways to prove Pythagoras' Theorem.
your telephone number is the sum of two prime numbers.
you have calculated that the World Series actually diverges.
you are sure that differential equations are a very useful tool.
you comment to your wife that her straight hair is nice and parallel.
when you say to a car dealer "I'll take the red car or the blue one" you must add "but not both of them."
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How many mathematicians does it take to change a light bulb?
None. It's left to the reader as an exercise.
None. The answer is intuitively obvious.
One. He gives it to four programmers, thereby simplifying the problem to a previous question.
How many numerical analysts does it take to change a light bulb?
3.9967 (after six iterations).
How many mathematical logicians does it take to change a light bulb?
None. They can't do it, but they can easily prove that it can be done.
How many classical geometers does it take to change a light bulb?
None. You can't do it with a straight edge and a compass.
How many analysts does it take to change a light bulb?
Three. One to prove existence, one to prove uniqueness and one to derive a nonconstructive algorithm to do it.
How many number theorists does it take to change a light bulb?
I don't know the exact number, but I am sure it must be some rather elegant prime.
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