How did the mathematician cure his constipation? He worked it out with pencil and sliderule. The gigantic computer took up a whole wall, dwarfing the two mathematicians standing before it. After much flashing and humming a sliver of paper emerged from the vitals of the machine. One mathematician, after studying it gravely, turned to the other and said with awe, "Do you realize that it would take four hundred ordinary mathematicians a hundred years of calculations to make a mistake this big?" A mathematician is a person who says that, when 3 people are supposed to be in a room but 5 came out, 2 have to go in so the room gets empty... Did you hear about the statistician who invented a device to measure the weight of trees? It's referred to as the log scale. Did you hear about the statistician who took the Dale Carnegie course? He improved his confidence from .95 to .99. Why don't statisticians like to model new clothes? Lack of fit. Did you hear about the statistician who was thrown in jail? 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. The last few available graves in a cemetary are called residual plots. Old statisticians never die, they just undergo a transformation. How do you tell one bathroom full of statisticians from another? Check the p-value. Did you hear about the statistician who made a career change and became an surgeon specializing in ob/gyn? 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. 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. Why is the number 10 afraid of seven? because seven ate nine. We use epsilons and deltas in mathematics because mathematicians tend to make errors. What's big, grey, and proves the uncountability of the reals? Cantor's Diagonal Elephant! How can you tell that Harvard was layed out by a mathematician? The div school [divinity school] is right next to the grad school. The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake, when it became known as SPLAC. SPLAC? Stanford Piecewise Linear Accelerator. How many topologists does it take to change a light bulb? It really doesn't matter, since they'd rather knot. Problem: To Catch a Deer in the woods 1. Mathematical Methods ------------------------ 1.1 The Hilbert (axiomatic) method ---------------------------------- We place a locked cage onto a given point in the woods. After that we introduce the following logical system: Axiom 1: The set of deers in the woods is not empty. Axiom 2: If there exists a deer in the woods, then there exists a deer in the cage. Procedure: If P is a theorem, and if the following is holds: "P implies Q", then Q is a theorem. Theorem 1: There exists a deer in the cage. 1.2 The geometrical inversion method ------------------------------------ We place a spherical cage in the woods, enter it and lock it from inside. We then perform an inversion with respect to the cage. Then the deer is inside the cage, and we are outside. 1.3 The projective geometry method ---------------------------------- Without loss of generality we can view the woods as a plane surface. We project the surface onto a line and afterwards the line onto an interior point of the cage. Thereby the deer is mapped onto that same point. 1.4 The Bolzano-Weierstrass method ---------------------------------- Divide the woods by a line running from north to south. The deer is then either in the eastern or in the western part. Lets assume it is in the eastern part. Divide this part by a line running from east to west. The deer is either in the northern or in the southern part. Lets assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the deer is caged into a fence of arbitrarily small diameter. 1.5 The set theoretical method ------------------------------ We observe that the woods is a separable space. It therefore contains an enumerable dense set of points which constitutes a sequence with the deer as its limit. We silently approach the deer in this sequence, carrying the proper equipment with us. 1.6 The Peano method -------------------- In the usual way construct a curve containing every point in the woods. It has been proven [1] that such a curve can be traversed in arbitrarily short time. Now we traverse the curve, carrying a rifle, in a time less than what it takes the deer to move a distance equal to its own length. 1.7 A topological method ------------------------ We observe that the deer possesses the topological gender of a torus. We embed the woods in a four dimensional space. Then it is possible to apply a deformation [2] of such a kind that the deer when returning to the three dimensional space is all tied up in itself. It is then completely helpless. 1.8 The Cauchy method --------------------- We examine a deer-valued function f(z). Be \zeta the cage. Consider the integral 1 [ f(z) ------- I --------- dz 2 \pi i ] z - \zeta C where C represents the boundary of the woods. Its value is f(zeta), i.e. there is a deer in the cage [3]. 1.9 The Wiener-Tauber method ---------------------------- We obtain a tame deer, D_0, from the class D(-\infinity,\infinity), whose fourier transform vanishes nowhere. We put this deer somewhere in the woods. D_0 then converges toward our cage. According to the general Wiener-Tauber theorem [4] every other deer D will converge toward the same cage. (Alternatively we can approximate D arbitrarily close by translating D_0 through the woods [5].) 2 Theoretical Physics Methods ----------------------------- 2.1 The Dirac method -------------------- We assert that wild deers can ipso facto not be observed in the woods. Therefore, if there are any deers at all in the woods, they are tame. We leave catching a tame deer as an exercise to the reader. 2.2 The Schroedinger method --------------------------- At every instant there is a non-zero probability of the deer being in the cage. Sit and wait. 2.3 The nuclear physics method ------------------------------ Insert a tame deer into the cage and apply a Majorana exchange operator [6] on it and a wild deer. As a variant let us assume that we would like to catch (for argument's sake) a male deer. We insert a tame female deer into the cage and apply the Heisenberg exchange operator [7], exchanging spins. 2.4 A relativistic method ------------------------- All over the woods we distribute deer bait containing large amounts of the companion star of Sirius. After enough of the bait has been eaten we send a beam of light through the woods. This will curl around the deer so it gets all confused and can be approached without being alerted to our presence. 2.5 The Newton method --------------------- Neglect friction and the deer and the cage will attract each other. 3 Experimental Physics Methods ------------------------------ 3.1 The thermodynamics method ----------------------------- We construct a semi-permeable membrane which lets everything but deers pass through. This we drag through the woods. 3.2 The atomic fission method ----------------------------- We irradiate the woods with slow neutrons. The deer becomes radioactive and starts to disintegrate. Once the disintegration process is progressed far enough the deer will be unable to resist. [1] After Hilbert, cf. E. W. Hobson, "The Theory of Functions of a Real Variable and the Theory of Fourier's Series" (1927), vol. 1, pp 456-457 [2] H. Seifert and W. Threlfall, "Lehrbuch der Topologie" (1934), pp 2-3 [3] According to the Picard theorem (W. F. Osgood, Lehrbuch derFunktionentheorie, vol 1 (1928), p 178) it is possible to catch every deer except for at most one. [4] N. Wiener, "The Fourier Integral and Certain of itsl Applications" (1933), pp 73-74 [5] N. Wiener, ibid, p 89 [6] cf e.g. H. A. Bethe and R. F. Bacher, "Reviews of Modern Physics", 8 (1936), pp 82-229, esp. pp 106-107 [7] ibid " The Saga of Polly Nomial Once upon a time, pretty Polly Nomial was skipping through a field of vectors when she came to the edge of a singularly large matrix. Now Polly was convergent, and her mother had made it an absolute condition that she never entered such an array without her brackets on. But Polly had changed her variables that morning and had been feeling particularly badly behaved, she ignored her mothers's condition on the grounds that it was insufficient, and made her way in among the complex elements. Rows and columns enveloped her on all sides. Tangents approached her surface. She grew tensor and tensor. Quite suddenly, three branches of a hyperbola touched her at a single point, she oscillated wildly and lost all sense of directrix. She tripped over a square root protruding from the erf, and tumbled headlong down a steep gradient. When she was once again in possesion of her variables, she found herself apparently in a non-euclidean space. She was being watched, however: that smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. Was she convergent? He wondered. He decided to integrate improperly at once. Hearing an improper fraction behind her, Polly rotated and saw Curly approaching with his power series extrapolated. She could tell at once from his degenerate conic and his dissipative terms that he was bent to no good. "Eureka!" she gasped. "Ho, ho," said our operator. "What a symetric little asymptote you have. I bet your angles are just dripping with secs." "Stay away from me!" she said. "I haven't got my brackets on." "Calm yourself, my dear," he said. "Your fears are purely imaginary." "I, I," she thought, "Maybe he's not normal..Maybe he's even a homomorphism." "What order are you?" the brute demanded. "Seventeen," she replied. Curly leered. "Enough of this idle chatter. Lets go to a decimal place I know, and I'll take you to the limit." "Never!" she gasped. "Arcsinh!!!" He swore the vilest oath he knew. Coshing her over the coefficient with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing out her points of inflection. Poor Polly. She could feel his hand tending towards her asymptotic limit. The algorithmic method was now her only hope. Her convergence would soon be gone forever. Curly's radius squared itself. Polly's loci quivered. He intergrated by parts. He intergrated by partial fractions. The complex beast even went all the way around and did a contour intergration. Curly went on operating until he was completely and totally exhausted of all his primitive roots. When Polly arrived home that night, her mother noticed that she had been truncated in several places. But it was too late to differentiate now. Nine transformations later, she went to L'Hopital and generated a small but pathological function which left zeros and residues all over the place and drove poor Polly to deviation. The moral of this story is: If you want to keep your expressions convergent, keep them well differentiated from complex operators. A mathematician is showing a new proof he came up with to a large group of peers. After he's gone through most of it, one of the mathematicians says "Wait! That's not true. I have a counter-example!" He replies, "That's okay. I have two proofs." Pythagoras. Now there's a guy you don't hear mentioned very much outside of a triangle context. Most people don't realize there was a lot more to this ancient Greek mathematician than just his famous theorem. Let's have a look at his achievements. Pythagoras' Theorem: The sum of the squares of the two legs of a right triangle is equal to the square of its hypotenuse. Pythagoras' Disclaimer: I know about hypotenuses, not hippopotamuses. Please, direct your hippo questions elsewhere. Pythagoras' Pick-Up Line: Hi, my name's Pythagoras. Perhaps you've heard of my theorem? Pythagoras' Secret Fear: Women don't dig triangle men. Pythagoras' Call To His Agent: I know, I know, the money's running a little thin. But I'm working on something that'll make us a fortune. Pythagoras' Witty Observation: Ever notice how, when you're heading out for a night of Dionysian theatre, you always end up wearing the same toga as your buddy? Ha ha! Ain't _that_ the truth! Pythagoras' Other Call To His Agent: What do you mean it tanked? OK, OK, keep your sandals on, I've got a sure thing coming down the pike. Pythagoras' Theorem II: The Revenge: If you square one leg of a right triangle only days before its retirement, the other leg will go on a one-leg quest for vengeance. Pythagoras' Third Call To His Agent: Oh, sure, they liked that, they love the triangle stuff, bunch of cretins. They want triangles, I'll give 'em triangles. Pythagoras' Theorem, Live Version: What does the sum of the squares of the two legs of a right triangle equal? I can't hear you? Yeah! The square of the hypotenuse! Thank you, goodnight! Pythagoras' Final Call To His Agent: No, Jerry, I'm through with the triangle stuff. You hear me? Uh huh? Go ahead and quit then, I don't _need_ an agent! Pythagoras' Not-Very Useful Rectangle Theorem: The sides of a rectangle are the part which you can find out the outside of the rectangle, whereas the middle part is the inside part of the rectangle. Pythagoras' Corollary to his Rectangle Theorem: That also goes for squares and other shapes. Pythagoras' Lame Statement of the Obvious: Um, sometimes it's nice to lie down and have a little nap. Pythagoras' Helpful Mnemonic Spelling Aid: When two vowels go walkin', the first one does the talkin'! Pythagoras' Complaint: Pi! Pi! it's all pi these days. Who cares?! Why compute the area of a circle?! Triangles are what count! Call me a has-been, will they! Hey, Uzo boy, bring us another round! Pythagoras' Theorem Revisited The Triangle Master's Great Theorem, Digitally Remastered So You Can Enjoy It Again and Again: The sum of the squares of the two legs of a triangle _still_ equals the square of its hypotenuse. Hey hey! Pythagoras' Touching Farewell Speech: In my life I may have had some bad breaks, but I consider myself the luckiest man in the world. Geometry has been very good to me. God bless, goodnight, and may all your hypotenuses be equal to the square root of the sums of the squares of the legs of your right triangles. A group of scientists were doing an investigation into problem-solving techniques, and constructed an experiment involving a physicist, an engineer, and a mathematician. The experimental apparatus consisted of a water spigot and two identical pails, one of which was fastened to the ground ten feet from the spigot. Each of the subjects was given the second pail, empty, and told to fill the pail on the ground. The physicist was the first subject: he carried his pail to the spigot, filled it there, carried it full of water to the pail on the ground, and poured the water into it. Standing back, he declared, "There: I have solved the problem." The engineer and the mathematician each approached the problem similarly. Upon finishing, the engineer noted that the solution was exact, since the volumes of the pails were equal. The mathematician merely noted that he had proven that a solution exists. Now, the experimenters altered the parameters of the task a bit: the pail on the ground was still empty, but the subjects were presented with a pail that was already half-filled with water. The physicist immediately carried his pail over to the one on the ground, emptied the water into it, went back to the spigot, *filled* the pail, and finally emptied the entire contents into the pail on the ground, overflowing it and spilling some of the water. Upon finishing, he commented that the problem should have been better stated. The engineer, in turn, thought for some time before going into action. He then took his half-filled pail to the spigot, filled it to the brim, and filled the pail on the ground from it. Again he noted that the problem had an exact solution, which of course he had found. The mathematician thought for a long time before stirring. At last he stood up, emptied his pail onto the ground, and declared, "The problem has been reduced to one already solved." First of all let me make it clear that I have nothing against contravariant functors. Some of my best friends are cohomology theories! But now you aren't supposed to call them contravariant anymore. It's Algebraically Correct to call them 'differently arrowed'!! In the same way that transcendental numbers are polynomially challenged? Manifolds are personifolds (humanifolds). Neighborhoods are neighbor victims of society. It's the Asian Remainder Theorem. It isn't PC to use "singularity" - the function is "convergently challenged" there. Here are some phrases used to remember SIN, COS, and TAN. (SIN = Opposite/Hypotenuse, COS = Adjacent/H, TAN = O/A). units and dimensions 2 monograms = 1 diagram 8 nickles = 2 paradigms 2 wharves = 1 paradox 10E5 bicycles = 2 megacycles 1 unit of suspense in an Agatha Christie novel = 1 whod unit Three Laws of Thermodynamics (paraphrased): First Law: You can't get anything without working for it. Second Law: The most you can accomplish by work is to break even. Third Law: You can't break even. A husband and wife, both statisticians, had the misfortune of passing away within a day of one another. They had always planned to be buried side by side. Unfortunately, the funeral home got them mixed up with another husband and wife with similar wishes. This became known as the first case of split-plot confounding. An engineer, a physicist and a mathematicians have to build a fence around a flock of sheep, using as little material as possible. The engineer forms the flock into a circular shape and constructs a fence around it. The physicist builds a fence with an infinite diameter and pulls it together until it fits around the flock. The mathematicians thinks for a while, then builds a fence around himself and defines himself as being outside. Mad Mathematicians News Item (June 23)-Mathematicians worldwide were excited and pleased today by the announcement that Princeton University professor Andrew Wiles had finally proved Fermat's Last Theorem, a 356-year-old problem said to be the most famous in the field. Yes, admittedly, there was rioting and vandalism last week during the celebration. A few bookstores had windows smashed and shelves stripped, and vacant lots glowed with burning piles of old dissertations. But overall we can feel relief that it was nothing- nothing- compared to the outbreak of exuberant thuggery that occurred in 1984 after Louis deBranges finally proved the Bieberbach Conjecture. "Math hooligans are the worst," said a Chicago Police Department spokesman. "But the city learned from the Bieberbach riots. We were ready for them this time." When word hit Wednesday that Fermat's Last Theorem had fallen, a massive show of force from law enforcement at universities all around the country headed off a repeat of the festive looting sprees that have become the traditional accompaniment to triumphant breakthroughs in higher mathematics. Mounted police throughout Hyde Park kept crowds of delirious wizards at the University of Chicago from tipping cars over on the midway as they first did in 1976 when Wolfgang Hakel and Kenneth Appel cracked the long-vexing Four-Color Problem. Incidents of textbook-throwing and citizens being pulled from their cars and humiliated with difficult story problems last week were described by the university's math department chairman Bob Zimmer as "isolated." Zimmer said, "Most of the celebrations were orderly and peaceful, But there will always be a few-usually graduate students-who use any excuse to cause trouble and steal. These are not true fans of Andrew Wiles." Wiles himself pleaded for calm even as he offered up the long elusive proof that there is no solution to the equation x^n + y^n = z^n when n is a whole number greater than two, as Pierre de Fermat first proposed in the 17th Century. "Party hard but party safe," he said, echoing the phrase he had repeated often in interviews with scholarly journals as he came closer and closer to completing his proof. Some authorities tried to blame the disorder on the provocative taunting of Japanese mathematician Yoichi Miyaoka. Miyaoka thought he had proved Fermat's Last Theorem in 1988, but his claims did not bear up under scrutiny of professional referees, leading some to suspect that the fix was in. And ever since, as Wiles chipped away steadily at the Fermat problem, Miyaoka scoffed that there would be no reason to board up windows near universities any time soon; that God wanted Miyaoka to prove it. In a peculiar sidelight, Miyaoka recently took the trouble to secure a U.S. trademark on the equation "x^n + y^n = z^n" as well as on the now-ubiquitous expression, "Take that, Fermat!" Ironically, in defeat, he stands to make a good deal of money on cap and T-shirt sales. This was no walk-in-the-park proof for Wiles. He was dogged, in the early going, by sniping publicity that claimed he was seen puttering late one night doing set theory in a New Jersey library when he either should have been sleeping, critics said, or focusing on arithmetic algebraic geometry for the proving work ahead. "Set theory is my hobby, it helps me relax," was his angry explanation. The next night, he channeled his fury and came up with five critical steps in his proof. Not a record, but close. There was talk that he thought he could do it all by himself, especially when he candidly referred to University of California mathematician Kenneth Ribet as part of his "supporting cast," when most people in the field knew that without Ribet's 1986 proof definitively linking the Taniyama Conjecture to Fermat's Last Theorem, Wiles would be just another frustrated guy in a tweed jacket teaching calculus to freshmen. His travails made the ultimate victory that much more explosive for math buffs. When the news arrived, many were already wired from caffeine comsumed at daily colloquial teas, and they took to the streets en masse shouting, "Obvious! Yessss! It was obvious!" The law cannot hope to stop such enthusiasm, only to control it. Still, one has to wonder what the connection is between wanton pillaging and a mathematical proof, no matter how long-awaited and subtle. The Victory Over Fermat rally, held on a cloudless day in front of a crowd of 30,000 (police estimate: 150,000) was pleasantly peaceful. Signs unfurled in the audience proclaimed Wiles the greatest mathematician of all time, though partisans of Euclid, Descartes, Newton and C.F. Gauss and others argued the point vehemently. A warmup act, The Supertheorists, delighted the crowd with a ragged song, "It Was Never Less Than Probable, My Friend," which included such gloating, barbed verses as- "I had my proof all ready/But then I did a choke-a/Made liberal assumptions/Hi! I'm Yoichi Miyaoka." In the speeches from the stage, there was talk of a dynasty, specifically that next year Wiles will crack the great unproven Riemann Hypothesis ("Rie-peat! Rie-peat!" the crowd cried), and after that the Prime-Pair Problem, the Goldbach Conjecture (Minimum Goldbach," said one T-shirt) and so on. They couldn't just let him enjoy his proof. Not even for one day. Math people. Go figure 'em. When people talk statistics, they always talk about averages. Average this, and average that. The thing to remember is that the average is exactly halfway between the best and the worst. It is the best of the worst and the worst of the best. So a mathematician, an engineer, and a physicist are out hunting together. They spy a deer in the woods. The physicist calculates the velocity of the deer and the effect of gravity on the bullet, aims his rifle and fires. Alas, he misses; the bullet passes three feet behind the deer. The deer bolts some yards, but comes to a halt, still within sight of the trio. "Shame you missed," comments the engineer, "but of course with an ordinary gun, one would expect that." He then levels his special deer-hunting gun, which he rigged together from an ordinary rifle, a sextant, a compass, a barometer, and a bunch of flashing lights which don't do anything but impress onlookers, and fires. Alas, his bullet passes three feet in front of the deer, who by this time wises up and vanishes for good. "Well," says the physicist, "your contraption didn't get it either." "What do you mean?" pipes up the mathematician. "Between the two of you, that was a perfect shot!" The difference between an Engineer and a Mathematician: The Engineer walks in her office and finds her trash can on fire. She gets the fire extinguisher and puts out the fire. The Mathematician walks in his office and finds his trash can on fire. He gets the fire extinguisher and puts out the fire. The following day: The Engineer walks in her office and finds the trash can on fire on top of her desk. She gets the fire extinguisher and put out the fire. The Mathematician walks in his office and finds the trash can on fire on top of his desk. He takes the trash can and puts it on the floor. He has reduced the problem to a previously solved state. Too solve it again would be redundant. A businessman needed to employ a quantitative type person. He wasn't sure if he should get a mathematician, an engineer, or an applied mathematician. As it happened, all the applicants were male. The businessman devised a test. The mathematician came first. Miss How, the administrative assistant took him into the hall. At the end of the hall, lounging on a couch, was a beautiful woman. Miss How said, "You may only go half the distance at a time. When you reach the end, you may kiss our model." The mathmatician explained how he would never get there in a finite number of iterations and politely excused himself. Then came the engineer. He quickly bounded halfway down the hall, then halfway again, and so on. Soon he declared he was well within accepted error tolerance and grabbed the beautiful woman and kissed her. Finally it was the applied mathematician's turn. Miss How explained the rules. The applied mathematician listened politely, then grabbed Miss How and gave her a big smooch. "What was that about?" she cried. "Well, you see I'm an applied mathematician. If I can't solve the problem, I change it!" Engineers think that equations approximate the real world. Physicists think that the real world approximates equations. Mathematicians are unable to make the connection ... If mathematicians are neutered, they can't multiply. |