By studying the masters, not their pupils N.H. Abel when asked how he managed to forge ahead so rapidly to the front rank according to E.T.Bell, Men of Mathematics Half the battle in mathematics is the invention of a good notation. P.S. Laplace This is the can opener that can open any can that can be opened by can openers. radio commercial L'abus des abstractions, l'abus des m\'etaphores, tel a \'et\'e, tel est encore le p\'eril de nos \'etudes. Nous avons vu les langues trait\'ees d'\^etres vivants: on nous a dit que les mots naissaient, se livraient des combats, se propagaient et mouraient. Michel Br\'eal, 1897 Essai de S\'emantique (Science des significations) One of the chief services which mathematics has rendered the human race is to put `common sense' where it belongs, on the topmost shelf next to the dusty canister labeled `discarded nonsense'. E.T. Bell, in Mathematics, Queen & Servant of Science (p. 18) Mathematics in general is fundamentally the science of self-evident things. F. Klein Mathematics is nothing more than a game played according to certain simple rules with meaningless marks on paper. D. Hilbert In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven. E. Everett (19 Nov 1863) Math = try to discover truths (Platonist) | enjoy the beauty of the structures and their relations (formalist) E.C.R. Hehner I want electricity to become so cheap that only the rich can afford candles. Edison, when asked about the future of the lightbulb To a great mind, nothing is minor. Sherlock Holmes The word "theoretical" has many connotations, but often it is simply taken to mean "impractical". As a theoretical computer scientist I do not feel that my work is impractical --on the contrary, I am convinced that practice without theory is bad practice-- but it has surprised me how difficult it is to persuade students to this view. Roland Backhouse ? Een buffer heeft de juiste grootte wanneer hij gemiddeld half vol is. Kruseman Aretz Let us have wine and women, mirth and laughter, Sermons and soda-water the day after. Byron (1788-1824) One fact which became obvious was that, although the simulator was found by most users to be of great benefit, it did not turn poor designers into good ones. Derrick Morris, Roland N. Ibbett in: The MU5 Computer System When a programming team are given a self-extensible language, some of the less desirable traits of human nature begin to show through. For example, the most prolific producers of new statements do not always make the wisest choice about what is needed. Nor are they over conscientious about documentation. Thus the size of the system can mushroom, and its operation and use moves into the folklore of the project. In a fairly short time the group as a whole can become separated from the reality of the hardware by a large `fuzzy' layer of software, and the efficiency suffers. Derrick Morris, Roland N. Ibbett in: The MU5 Computer System Graduates of one-week first-aid courses do not normally think they are ready to perform brain surgery, one-week programmers, however, seem willing to try tasks of equivalent difficulty. The patient invariably dies. H. Whitfield We don't understand brains /\ we don't understand neural networks => brains = neural networks Chuck Seitz They have been at a great feast of languages, and stolen the scraps. Shakespeare: Love's Labours's Lost. Never be negative. note on David Gries's refrigerator The purpose of computing is insight, not numbers. R.W. Hamming Engineers think that theory approximates reality. Physicists think that reality approximates theory. Mathematicians never make the connection. anon When FORTRAN has been called an infantile disorder, full PL/I, with its growth characteristics of a dangerous tumor, could turn out to be a fatal disease. Edsger W. Dijkstra in The Humble Programmer Turing Award Lecture, 14 August 1972 Switches get faster 15% every year, wires don't. Bill Dally, DARPA meeting September 1992 Reality is 50-100 million triangles. Henry Fuchs, DARPA meeting September 1992, when explaining how much computational power is needed for realistic images Give us the tools, and we will finish the job. Winston Leonard Spencer Churchill Radio Broadcast 9 Feb 1941 (addressing President Roosevelt) "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 to the bonds of Hell." St. Augustine (but see next) mathematicus: 1. mathematician 2. astrologer (The Oxford Latin Dictionary) He who hasn't hacked assembly language as a youth has no heart. He who does so as an adult has no brain. John Moore Gedanken sind nicht stets parat, man schreibt auch, wenn man keine hat. Wilhelm Busch Son, all the pretty, intelligent, healthy # Ken Johnson, AIAI, young women are taken. It's a basic rule of # 80 South Bridge, Edinburgh the universe, and if you don't like it, go # Tel 031-650 2756 somewhere else. -- my dad 1906-1992 # Fax 031-650 6513 We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough. Niels Bohr (about Wolfgang Pauli's theory) Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different. Johann Wolfgang von Goethe C++ is the best example of second-system| Henry Spencer @ U of Toronto Zoology effect since OS/360. | henry@zoo.toronto.edu utzoo!henry We build information systems like the Wright Brothers built airplanes ... build the whole thing, push it off the cliff, let it crash, and start over again. NATO Scientific Affairs Division This isn't right. This isn't even wrong. Wolfgang Pauli Einstein was a genius: Head in the clouds, feet on the ground. But those of us who are not as tall, have to make a choice. Dick Feynman If all else fails, show pretty pictures and animated videos, and don't talk about performance David Bailey For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled. Richard P. Feynman The fastest chemical reactions in the brain take 1 millionth of a second, so 1 MIPS would be an absolute Peak figure, sustained performance is much lower. Also with regards to Memory, it's actually 11.6GB in 1.25kg with a loss of 3KB per day after the age of 18. Gunter Ahrendt Entia non sunt multiplicanda praeter necessitatem. Occam I love mathematics, but it's mathematicians I cannot stand. Edsger W. Dijkstra (in EWD 1151) I think that the historical importance of LISP is that it was the first language to provide `garbage-collected' heap storage. For the development of functional programming, however, I feel that the contribution of LISP has been a negative one. My suspicion is that the success of LISP set back the development of a properly functional style of programming by at least ten years. D.A. Turner see C.A.R. Hoare, J.C. Shepherdson Mathematical Logic and Programming Languages page 53 Given that true intellectual and emotional compatibility are at the very least difficult if not impossible to come by; we could always opt for the more temporal gratification of sheer physical attraction. That would not make you a shallow person, would it? Lyle Lovett By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. A.N. Whitehead By the aid of symbolism we can make transitions in reasoning almost mechanically by the eye, which would otherwise call into play the higher faculties of the brain. It is a profoundly erroneous truism, repeated by copy books and by eminent people when they make speeches, that we cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. A.N. Whitehead There are three kinds of mathematicians: those who can count, and those who cannot count. Anon The cheapest, fastest, and most reliable components of a computer system are those aren't there. Gordon Bell At first I hoped that such a technically unsound project would collapse but I soon realized it was doomed to success. Almost anything in software can be implemented, sold, and even used given enough determination. There is nothing a mere scientist can say that will stand against the flood of a hundred million dollars. But there is one quality that cannot be purchased in this way --- and that is reliability. The price of reliability is the pursuit of the utmost simplicity. It is a price which the very rich find most hard to pay. C.A.R. Hoare in The Emperor's Old Clothes, Turing Award Lecture delivered 27 October 1980 Confucious, Master of Wisdom, teaches us, "Look before you leap." His disciple, Leng-Tzu, teaches, "He who hesitates is lost." Confucious whispers "Don't put all of your eggs in one basket." Leng-Tzu reminds us "Nothing ventured, nothing gained." Confucious says "Absence makes the heart grow fonder." Leng-Tzu says "Out of sight, out of mind." Confucious says "The best things in life are worth waiting for." Leng-Tzu says "Live each day as if it were your last." Confucious says "Don't judge a man until you have walked a mile in his shoes." Leng-Tzu teaches "Familiarity breeds contempt." Confucious teaches "Birds of a feather flock together." Leng-Tzu says "Opposites attract." Confucious insists "It's the thought that counts." Leng-Tzu begs to differ "The road to hell is paved with good intentions." Confucious states "Actions speak louder than words." Leng-Tzu corrects "The pen is mightier than the sword." Confucious says "Good things come in small packages." Leng-Tzu points out "The bigger the better." Confucious notes "A carpenter is only as good as his tools." Leng-Tzu admonishes "'Tis a poor workman who blames his tools." Confucious points out "The best things in life are free." Leng-Tzu responds "You get what you pay for." Confucious explains "Variety is the spice of life." Leng-Tzu maintains "The cat that chases two mice catches none." Confucious teaches "If at first you dont succeed, try, try again." Leng-Tzu replies "Quit while you're ahead." Confucious preaches "Hope springs eternal." Leng-Tzu prescribes "Dont count your chickens before they are hatched." Confucious suggests "A stitch in time saves nine." Leng-Tzu directs "If it ain't broke, don't fix it." Confucious instructs "The early bird catches the worm." Leng-Tzu reciprocates "Better late than never" Confucious observes "Idle hands are the devil's playground." Leng-Tzu rebounds "All work and no play makes Jack a dull boy." Confucious reminds "The more, the merrier." Leng-Tzu elaborates "Two's company, three's a crowd." Confucious claims "Take care of the pennies and the pounds will take care of themselves." Leng-Tzu advocates "Penny wise, pound foolish." Confucious advises "Do unto others as you would have them do unto you." Leng-Tzu expounds "Eye for an eye, tooth for a tooth." Confucious expresses "It's never too late too learn!" Leng-Tzu interjects "You can't teach an old dog new tricks!" Confucious pontificates "What you leave alone today will come back to haunt you tomorrow." Leng-Tzu asserts "Let sleeping dogs lie." Confucious states "Strike while the iron is hot." Leng-Tzu counters "Haste makes waste." Confucious relates "Knowledge is power." Leng-Tzu rebutts "Ignorance is bliss." Confucious intones "A man of peace is a man for all seasons." Leng-Tzu retorts "Might makes right." Confucious opines "Many hands make light work." Leng-Tzu disputes "Too many cooks spoil the broth." Confucious declares "Silence is golden." Leng-Tzu argues "The squeeky wheel gets the grease." Confucious establishes "Better safe than sorry." Leng-Tzu refutes "No guts, no glory." Confucious contends "A coward dies a thousand deaths, a hero dies but once." Leng-Tzu contravenes "He who fights and runs away lives to fight another day." Confucious entreats "Don't look a gift horse in the mouth." Leng-Tzu dissents "Beware of Greeks bearing gifts." Confucious warns "Fat paunches have lean pates." Leng-Tzu rebukes "Eat, drink, and be merry." Confucious insists "Don't judge a book by its cover." Leng-Tzu contradicts "Where there's smoke, there's fire." Confucious mandates "A donkey raised among horses is still a donkey." Leng-Tzu wrangles "Even a hawk is an eagle among crows." Confucious bellows "If you can't beat 'em, join 'em." Leng-Tzu quarrels "Dont let the bastards wear you down." Confucious demands "A cut of the tongue can wound deeper than swords." Leng-Tzu shouts "Sticks and stones may break my bones but names will never hurt me." ================================================================ THE PROGRAMMER'S QUICK GUIDE TO THE LANGUAGES The proliferation of modern programming languages (all of which seem to have stolen countless features from one another) sometimes makes it difficult to remember what language you're currently using. This handy reference is offered as a public service to help programmers who find themselves in such a dilemma. TASK: Shoot yourself in the foot. C: You shoot yourself in the foot. C++: You accidentally create a dozen instances of yourself and shoot them all in the foot. Providing emergency medical assistance is impossible since you can't tell which are bitwise copies and which are just pointing at others and saying, "That's me, over there." FORTRAN: You shoot yourself in each toe, iteratively, until you run out of toes, then you read in the next foot and repeat. If you run out of bullets, you continue with the attempts to shoot yourself anyways because you have no exception-handling capability. Pascal: The compiler won't let you shoot yourself in the foot. Ada: After correctly packing your foot, you attempt to concurrently load the gun, pull the trigger, scream, and shoot yourself in the foot. When you try, however, you discover you can't because your foot is of the wrong type. COBOL: Using a COLT 45 HANDGUN, AIM gun at LEG.FOOT, THEN place ARM.HAND.FINGER on HANDGUN.TRIGGER and SQUEEZE. THEN return HANDGUN to HOLSTER. CHECK whether shoelace needs to be re-tied. LISP: You shoot yourself in the appendage which holds the gun with which you shoot yourself in the appendage which holds the gun with which you shoot yourself in the appendage which holds the gun with which you shoot yourself in the appendage which holds the gun with which you shoot yourself in the appendage which holds the gun with which you shoot yourself in the appendage which holds... FORTH: Foot in yourself shoot. Prolog: You tell your program that you want to be shot in the foot. The program figures out how to do it, but the syntax doesn't permit it to explain it to you. BASIC: Shoot yourself in the foot with a water pistol. On large systems, continue until entire lower body is waterlogged. Visual Basic: You'll really only _appear_ to have shot yourself in the foot, but you'll have had so much fun doing it that you won't care. HyperTalk: Put the first bullet of gun into foot left of leg of you. Answer the result. Motif: You spend days writing a UIL description of your foot, the bullet, its trajectory, and the intricate scrollwork on the ivory handles of the gun. When you finally get around to pulling the trigger, the gun jams. APL: You shoot yourself in the foot, then spend all day figuring out how to do it in fewer characters. SNOBOL: If you succeed, shoot yourself in the left foot. If you fail, shoot yourself in the right foot. Unix: % ls foot.c foot.h foot.o toe.c toe.o % rm * .o rm:.o no such file or directory % ls % Concurrent Euclid: You shoot yourself in somebody else's foot. 370 JCL: You send your foot down to MIS and include a 400-page document explaining exactly how you want it to be shot. Three years later, your foot comes back deep-fried. Paradox: Not only can you shoot yourself in the foot, your users can, too. Access: You try to point the gun at your foot, but it shoots holes in all your Borland distribution diskettes instead. Revelation: You're sure you're going to be able to shoot yourself in the foot, just as soon as you figure out what all these nifty little bullet-thingies are for. Assembler: You try to shoot yourself in the foot, only to discover you must first invent the gun, the bullet, the trigger, and your foot. Modula2: After realizing that you can't actually accomplish anything in this language, you shoot yourself in the head. ================================================================ Here's a similar file that discusses various operating systems. The Task: GO TO THE STORE MS-DOS: You get in the car and try to remember where you put your keys. Windows: You get in the car and drive to the store very slowly, because attached to the back of the car is a freight train. Macintosh System 7: You get in the car to go to the store, and the car drives you to church. UNIX: You get in the car and type GREP STORE. After reaching speeds of 200 miles per hour en route, you arrive at the barber shop. Windows NT: You get in the car and write a letter that says "go to the store." Then you get out of the car and mail the letter to your dashboard. Taligent/Pink: You walk to the store with Ricardo Montalban, who tells you how wonderful it will be when he can fly you to the store in his Learjet. OS/2: After fueling up with 6000 gallons of gas, you get in the car and drive to the store with a motorcycle escort and a marching band in procession. Halfway there, the car blows up, killing everybody in town. S/36 SSP [mainframe, obv.]: You get in the car and drive to the store. Halfway there you run out of gas. While walking the rest of the way, you are run over by kids on mopeds. AS/400: An attendant locks you into the car and then drives you to the store, where you get to watch everybody else buy filet mignons. ================================================================ This might prove to be of interest in understanding the terminology used in research papers :: GLOSSARY FOR RESEARCH PAPERS: Strictly Speaking THEY WRITE THEY MEAN ---------- --------- It has long been known that... I haven't bothered to look up the original reference ...of great theoretical and practical ...interesting to me importance While it has not been possible to The experiments didn't work provide definite answers to these out, but I figured I could at questions. least get a publication out of it. The careful reader will observe : I don't know how to prove it: It can be trivially see and Q.E.D and if you can't figure it out.. boy..u'r pretty dumb then. The W-Pb system was chosen as It was the only one that especially suitable to show the worked.. predicted behavior... Three of the samples results were The results of the others chosen for detailed study... didn't make sense and were ignored.. Typical results are shown... The best results are shown... Although some detail has been lost It is impossible to tell from in reproduction, it is clear from the the micrograph original micrograph that... Presumably at longer times... I didn't take the time to find out The agreement with the predicted fair curve is excellent : good poor : satisfactory doubtful : fair imaginary ...as good as could be expected non-existent These results will be reported at I might get around to this a later date sometime The most reliable values are those He was a student of mine of Jones It is suggested that... It is believed that... I think... It may be that... It is generally believed that.... I have such a good objection to this answer that I shall now raise it. It is clear that much additional I don't understand it work will be required before a complete understanding... Unfortunately, a quantitative theory Neither does anybody else to account for these effects has not been formulated Correct within an order of magnitude Wrong It is to be hoped that this work This paper isn't very good but will stimulate further work in the field neither are any of the others on this miserable subject Thanks are due to Joe Glotz for Glotz did the work and Doe assistance with the experiments and explained what it meant. to John Doe for valuable discussions. ================================================================ The heaviest element known to science was discovered at the Lawrence Livermore National Laboratory. The element, tentatively named ADMINISTRATUM (Ad), has no protons or electrons. Thus it has the atomic number 0. It does, however, have one neutron, 75 associate neutrons, 125 deputy associate neutrons, and 111 assistant deputy neutrons. The 312 particles are held together in the nucleus by a force that involves the continuous exchange of meson-like particles called memons. Since it has no electrons, ADMINISTRATUM is inert. Nevertheless, it can be detected chemically because it seems to impede every reaction in which it takes part. According to Dr. Langour, one of the discoverers of the element, a very small amount of ADMINISTRATUM made one reaction that normally takes less than a second, take over four days to go to completion. ADMINISTRATUM has a half-life of approximately three years, at which time it does not actually decay. Instead, it undergoes an internal reorganization in which associates to the neutron, deputy associates to the neutron, and assistant deputy associate neutrons all exchange places. Some studies have indicated that the atomic mass actually increases after each reorganization. ================================================================ Math Riots Prove Fun Incalculable --------------------------------- The following column appeared in the Chicago Tribune, County edition Tuesday June 29 1993 page 2-1. .............................................................................. Math Riots Prove Fun Incalculable by Eric Zorn /begin italics/ 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 365-year-old problem said to be the most famous in the field. /end italics/ 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 over cars on the midway as they first did in 1976 when Wolfgang Haken 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 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 the 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 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 consumed at daily colloquial teas, and the 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 a 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 that after 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. ================================================================ Problem: To Catch a Lion in the Sahara Desert. 1. Mathematical Methods 1.1 The Hilbert (axiomatic) method We place a locked cage onto a given point in the desert. After that we introduce the following logical system: Axiom 1: The set of lions in the Sahara is not empty. Axiom 2: If there exists a lion in the Sahara, then there exists a lion in the cage. Procedure: If P is a theorem, and if the following holds: "P implies Q", then Q is a theorem. Theorem 1: There exists a lion in the cage. 1.2 The geometrical inversion method We place a spherical cage in the desert, enter it and lock it from inside. We then perform an inversion with respect to the cage. Then the lion is inside the cage, and we are outside. 1.3 The projective geometry method Without loss of generality, we can view the desert as a plane surface. We project the surface onto a line and afterwards the line onto an interiour point of the cage. Thereby the lion is mapped onto that same point. 1.4 The Bolzano-Weierstrass method Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part. Let's assume it is in the eastern part. Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Let's 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 lion is caged into a fence of arbitrarily small diameter. 1.5 The set theoretical method We observe that the desert is a separable space. It therefore contains an enumerable dense set of points which constitutes a sequence with the lion as its limit. We silently approach the lion 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 desert. It has been proven [1] that such a curve can be traversed in arbitrarily short time. Now we traverse the curve, carrying a spear, in a time less than what it takes the lion to move a distance equal to its own length. 1.7 A topological method We observe that the lion possesses the topological gender of a torus. We embed the desert in a four dimensional space. Then it is possible to apply a deformation [2] of such a kind that the lion 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 lion-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 desert. Its value is f(zeta), i.e. there is a lion in the cage [3]. 1.9 The Wiener-Tauber method We obtain a tame lion, L_0, from the class L(-\infinity,\infinity), whose Fourier transform vanishes nowhere. We put this lion somewhere in the desert. L_0 then converges toward our cage. According to the general Wiener-Tauner theorem [4] every other lion L will converge toward the same cage. (Alternatively we can approximate L arbitrarily close by translating L_0 through the desert [5].) 2 Theoretical Physics Methods 2.1 The Dirac method We assert that wild lions can ipso facto not be observed in the Sahara desert. Therefore, if there are any lions at all in the desert, they are tame. We leave catching a tame lion as an exercise to the reader. 2.2 The Schroedinger method At every instant there is a non-zero probability of the lion being in the cage. Sit and wait. 2.3 The Quantum Measurement Method We assume that the sex of the lion is _ab initio_ indeterminate. The wave function for the lion is hence a superposition of the gender eigenstate for a lion and that for a lioness. We lay these eigenstates out flat on the ground and orthogonal to each other. Since the (male) lion has a distinctive mane, the measurement of sex can safely be made from a distance, using binoculars. The lion then collapses into one of the eigenstates, which is rolled up and placed inside the cage. 2.4 The nuclear physics method Insert a tame lion into the cage and apply a Majorana exchange operator [6] on it and a wild lion. As a variant let us assume that we would like to catch (for argument's sake) a male lion. We insert a tame female lion into the cage and apply the Heisenberg exchange operator [7], exchanging spins. 2.5 A relativistic method All over the desert we distribute lion 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 desert. This will curl around the lion so it gets all confused and can be approached without danger. 3 Experimental Physics Methods 3.1 The thermodynamics method We construct a semi-permeable membrane which lets everything but lions pass through. This we drag across the desert. 3.2 The atomic fission method We irradiate the desert with slow neutrons. The lion becomes radioactive and starts to disintegrate. Once the disintegration process is progressed far enough the lion will be unable to resist. 3.3 The magneto-optical method We plant a large, lense shaped field with cat mint (nepeta cataria) such that its axis is parallel to the direction of the horizontal component of the earth's magnetic field. We put the cage in one of the field's foci. Throughout the desert we distribute large amounts of magnetized spinach (spinacia oleracea) which has, as everybody knows, a high iron content. The spinach is eaten by vegetarian desert inhabitants which in turn are eaten by the lions. Afterwards the lions are oriented parallel to the earth's magnetic field and the resulting lion beam is focussed on the cage by the cat mint lense. [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 der Funktionentheorie, vol 1 (1928), p 178) it is possible to catch every lion 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 -- 4 Contributions from Computer Science. 4.1 The search method We assume that the lion is most likely to be found in the direction to the north of the point where we are standing. Therefore the REAL problem we have is that of speed, since we are only using a PC to solve the problem. 4.2 The parallel search method. By using parallelism we will be able to search in the direction to the north much faster than earlier. 4.3 The Monte-Carlo method. We pick a random number indexing the space we search. By excluding neighboring points in the search, we can drastically reduce the number of points we need to consider. The lion will according to probability appear sooner or later. 4.4 The practical approach. We see a rabbit very close to us. Since it is already dead, it is particularly easy to catch. We therefore catch it and call it a lion. 4.5 The common language approach. If only everyone used ADA/Common Lisp/Prolog, this problem would be trivial to solve. 4.6 The standard approach. We know what a Lion is from ISO 4711/X.123. Since CCITT have specified a Lion to be a particular option of a cat we will have to wait for a harmonized standard to appear. $20,000,000 have been funded for initial investigations into this standard development. 4.7 Linear search. Stand in the top left hand corner of the Sahara Desert. Take one step east. Repeat until you have found the lion, or you reach the right hand edge. If you reach the right hand edge, take one step southwards, and proceed towards the left hand edge. When you finally reach the lion, put it in the cage. If the lion should happen to eat you before you manage to get it in the cage, press the reset button, and try again. 4.8 The Dijkstra approach: The way the problem reached me was: catch a wild lion in the Sahara Desert. Another way of stating the problem is: Axiom 1: Sahara elem deserts Axiom 2: Lion elem Sahara Axiom 3: NOT(Lion elem cage) We observe the following invariant: P1: C(L) v not(C(L)) where C(L) means: the value of "L" is in the cage. Establishing C initially is trivially accomplished with the statement ;cage := {} Note 0: This is easily implemented by opening the door to the cage and shaking out any lions that happen to be there initially. (End of note 0.) The obvious program structure is then: ;cage:={} ;do NOT (C(L)) -> ;"approach lion under invariance of P1" ;if P(L) -> ;"insert lion in cage" [] not P(L) -> ;skip ;fi ;od where P(L) means: the value of L is within arm's reach. Note 1: Axiom 2 esnures that the loop terminates. (End of note 1.) Exercise 0: Refine the step "Approach lion under invariance of P1". (End of exercise 0.) Note 2: The program is robust in the sense that it will lead to abortion if the value of L is "lioness". (End of note 2.) Remark 0: This may be a new sense of the word "robust" for you. (End of remark 0.) Note 3: From observation we can see that the above program leads to the desired goal. It goes without saying that we therefore do not have to run it. (End of note 3.) (End of approach.) ---------------------------------------------------- "We was heading' south with 456 - 32 loads and 10 empties and we had a meet order with 2 for 149 at 4:15. There was a dynamiter in drag and it looked like we was goin' to stab 149 about 15 but the bronco quit buckin' and the hogger wheeled them, so we got to 149 at 4:10. We went in on the cold end of the passin' track and closed the gate, drifted down to about ten lengths from the south target and grounded the kettle. The hoghead got down and went greasin' her ankles and the tallowpot was in the cab hustlin' diamonds. Cap was back in the crummie shufflin' tissues and the hind-shack was in the doghouse bullin' a Yahoo that was dead-headin' on a Zulu outfit. Me, I was sittin' on the pilot waitin' to twist the monkey's tail and head out on the main when 2 cleared. I seen her comin' through the cut and she was workin' steam and puttin' up a feather. Hogger cut off at the limit board and she come in greasin' at about 25. When she passed the target, the ponies split the switch and the took the siding. The hogger big-holed her but she jumped the rabbit and took the country. I seen a smash was comin' and joined the bird gang. I hit the cinders yellin' "Spread yer wings, fellers." and when they came together, I was on the other side of the main scratchin' gravel. I dodged into the palace and told the brass pounder to tickle the Supe and tell him we had two on one spot and there was Hell to pay." _________________________________________________________________ The above is an actual account of testimony from a trial handled on behalf of the AT&SF by Attorney Luther Hudson. ----------------------------------- (Now the same thing - annotated) "We was heading' south with 456 - Train number 456 32 loads and 10 empties and we had a meet order with 2 for 149 at 4:15. Meeting train #2 at milepost 149 at 4:15 There was a dynamiter in drag and it Dynamiter: car with defective brake gear that intermittently applies itself looked like we was goin' to stab 149 Stab 149: reach milepost 149 about 15 but the bronco quit buckin' about 15: 15 minutes past the hour; 4:15 bronco quit buckin': the dynamiter quit applying the brakes, so the train no longer "bucked" and the hogger wheeled them, so we got hogger: engineer wheeled them: made good time to 149 at 4:10. We went in on the cold end of the passin' track and cold end: facing point switch end closed the gate, drifted down to about closed the gate: aligned switch for main ten lengths from the south target and target: sign on switchstand grounded the kettle. The hoghead got grounded the kettle: stopped the engine hoghead: see hogger down and went greasin' her ankles and greasin' her ankles: oiling the many joints in the drive train and valve gear the tallowpot was in the cab hustlin' tallowpot: fireman hustlin' diamonds: shovelling coal diamonds. Cap was back in the crummie Cap: the conductor crummie: caboose shufflin' tissues and the hind-shack tissues: train orders; "flimsies" hind-shack: rear brakeman was in the doghouse bullin' a Yahoo doghouse: brakeman's shelter on back of tender Yahoo: hobo that was dead-headin' on a Zulu dead-headin': hitching a free ride; in general, anything (RR employees, engines, rolling stock) travelling in a non-revenue-generating fashion outfit. Me, I was sittin' on the Zulu outfit: ??? pilot waitin' to twist the monkey's pilot: front of the engine twist the monkey's tail: move the switchstand lever to through the switch tail and head out on the main when 2 cleared. I seen her comin' through the cut and she was workin' steam and puttin' workin' steam: the engine was working hard up a feather. Hogger cut off at the puttin' up a feather: engine sending up a plume of steam and smoke from the stack limit board and she come in greasin' limit board: speed limit sign (or yard limit?) greasin': coasting at about 25. When she passed the 25: 25 mph target, the ponies split the switch ponies: small wheels on the pilot truck split the switch: flanges of wheels force the switch points over and take the wrong branch of the switch and the took the siding. The hogger big-holed her but she jumped the big-holed her: applied emergency brakes rabbit and took the country. rabbit: synonym for frog; intersection of stock rails at a switch took the country: derailed I seen a smash was comin' and joined the bird gang. I hit the joined the bird gang: flee cinders yellin' "Spread yer wings, cinders: ballast, roadbed fellers." and when they came together, I was on the other side of came together: trains collided the main scratchin' gravel. I dodged scratchin' gravel: walking (or in this case, running) on the ballast into the palace and told the brass palace: depot, station pounder to tickle the Supe and tell brass pounder: telegraph operator him we had two on one spot and there tickle the Supe: notify the Superintendent was Hell to pay." two on one spot: two trains on the same track; collision ----------------------------------------------------------------------- from Digger O'Balls Funeral Parlor in Tombstone, Arizona: - you furnish the bones & we'll furnish the stones - why go around half dead when we can bury you for $49.50 - when you get shot and before you are stiff we'll be there in just a jiff - we'll be the last to let you down ================================================================ ... the reason why human understanding has been able to advance in the past, and may do so in the future, is that true insights are cumulative and retain their value regardless of what happens to their discoverers; while fads and stunts may bring an immediate profit to the impresarios, but lead nowhere in the long run, cancel each other out, and are dropped as soon as their promoters are no longer there (or have lost their power) to direct the show. Styanislav Andreski in Social Science as Sorcery But unfortunately, the contours of truth never coincide with the frontiers between embattled parties and cliques. Styanislav Andreski in Social Science as Sorcery