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COMIC SECTIONS The Book of Mathematical Jokes,

Humour , Wit and Wisdom i

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DESMOND MACHALE 1

BOOLE PRESS DUBLIN MCMXCI1 I

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CONTENTS

All rights reserved . No part of this publication may be reproduced , stored in a retrieval system or transmitted in any form or by any means, electronic , electro static , magnetic tape , mechanical , photocopying , recording or otherwise , without prior permission from the publishers.

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Boole Press, 26 Temple Lane, Dublin 2, Ireland Copyright © Boole Press First Edition 1993

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Hardback ISBN 1 85748-006-6 Paperback ISBN 1-85748-007-4

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Chapter 0

Introduction

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Chapter 1

Anecjokes

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Chapter 2

Predicting Counterexamples

Chapter 3

Limericks

Chapter 4

Howlers and Boners

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Chapter 5

Mathematics and Ethnic Jokes

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Chapter 6

A Short Dictionary of Mathematical Terms

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Chapter 7

Mathematical Graphiti

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Chapter 8

Some Reviews from the Journals

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Chapter 9

A Few Riddles

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Chapter 10

Mathematical Wit and Wisdom

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and Verse

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Chapter 11 Those Magnificent Men on their Ttiring Machines

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Chapter 12 The Final Examination

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INTRODUCTION

Much of this book is devoted to what I call “anecjokes”: that is, stories, jokes and anecdotes about mathematics and mathematicians, or jokes with a mathematical or logical content. In twenty years of teaching mathematics at various levels I have come to believe that such material is not necessarily flippant but in fact has a serious if not essential function in the study and understanding of mathematics. At the very least, it humanises mathematics and can persuade students that at least some mathematicians are human and that even the most abstruse and abstract mathematics is the product of human minds. Mathematics and humour are closely related and logic is undoubtedly a connecting link. Mathematics is supposedly ultra-logical; humour is often ultra-illogical. Mathematicians are considered good at analysing logical situations, and perhaps they are also more sensitive to the type of humour in which the laws of logic are flouted. When Bertrand Russell remarked, "Mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true,” he hit upon an insight whose illogicality appeals far more to the mathematician than to the layman. Einstein aimed at the same target with his observation, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality "

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In my experience, the level of appreciation of humour among mathematicians is very high indeed, at least as high as that of any other profession I know of. This is perhaps because mathematicians have to be able to face so many jokes at their expense, and mainly from other mathematicians. The absent-minded professor is of course a stock figure of humour, but he seems to be specially comical when it is mathematics that he professes. There may be a compliment wrapped up here, an unconscious recognition that it is the mathematician who reaches "the highest rung on the ladder of human thought ,” as Havelock Ellis claimed.

Humour is an excellent way of teaching certain areas of mathematics as long as it is not over used. Humour can inspire insight and aid the memory, and certain techniques and tricks in mathematical proofs are themselves almost jokes. The

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paradox is in some ways the true meeting point of mathematics and humour, and a teacher who presents a paradoxical proof that 2 = 1 to an elementary algebra class will find that a discussion and analysis of this joke will lead to more understanding of mathematics than several hours of axioms and lemmas

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Historically there have been several mathematical humorists, humorous math ematicians or people with a foot in both camps. We may cite Lewis Carroll, Stephen Leacock , Tom Lehrer, Martin Gardner, Charles Babbage , John Paulos, Leo Moser and Clerk Maxwell. One of my own favourites, Silvanus P. Thomp son, seems nearly forgotten , though his book Calculus Made Easy was reprinted for at least fifty years and is still a joy to the devotee. He had a slogan , “What one fool can do, another can,” which he claimed was an ancient Simian proverb. It has heartened many a youngster who might otherwise have given up dx and / in despair. A Fellow of the Royal Society, Thompson used humour as a weapon against professionalism in mathematics: “It may be confidently assumed that when this tractate falls into the hands of the professional mathematicians, they will ( if not too lazy) rise up as one man, and damn it as being a thoroughly bad book. Of that there can be, from their point of view, no possible manner of doubt whatever. It commits several most grievous and deplorable

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errors. First, it shows how ridiculously easy most of the operations of the calculus really are. Secondly, it gives away so many trade secrets.. . ” This opened , in the year 1910, a door of mathematical humour and wisdom which still lets in a refreshing breeze.

Is there a greater appreciation of humour among people in certain branches of mathematics? My personal belief is that discrete mathematics, algebra and combinatorics are more akin to humour than are analysis and continuous mathematics. I have absolutely no evidence for this assertion , and fully appreciate that anyone who would care to be inundated with counterexamples has only to make such a statement public. I am grateful to my friend Brendan O’Brien for the artistry used on the cartoons. In fairness to Brendan, I must admit that the choice of themes was mine alone.

Finally, a dedication: To all those mathematicians and students of mathematics worldwide who unwittingly made this book possible .

NOTE TO THE READER I am aware that this book contains a great many errors, mistakes, blunders and omissions which I have been too lazy to correct. I would be very grateful if readers would not write pointing these out to me.

Let us start this chapter with a little game which is not without its sardonic humour. The game is played by two players A and B and there is a neutral referee R . Each player is allowed to pick a number, suppose that A picks a and B picks b If a = b , the referee asks the players to choose new numbers until o 7^ b. If a > b , then A is allowed to kill B , while if b > a , B is allowed to kill A . However, and this is the catch, if a + 6 > 100, the referee R must kill both A and B. Although this game has no official name, it has been played on a worldwide scale. Some people call it the nuclear game. In a nice variation , neither A nor B knows that the value in question is 100.

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There once was a Red Indian chief who had three wives. They were allowed to choose bedding on which to sleep and each wife chose according to her taste. The first chose a luxurious hippopotamus skin , the second a lion skin and the third a tiger skin. In the fullness of time , the first wife brought forth triplets, the second wife had twins but the third wife could only manage a single baby. The moral of the story is that “The squaw on the hippopotamus is equal to the sum of the squaws on the other two hides”

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The calendar of the institution in which I have the honour to teach once printed the following item in the curriculum for the statistics section of biomathematics: 2 X -testes

Most mathematicians receive letters from time to time from harmless lunatics who claim they can trisect angles or square the circle. Sometimes the letters contain ingenious but utterly useless suggestions. A friend of mine, Ken Nicholls, reports that his father, a mathematics inspector, received a letter with the following gem: If every circle were given a straight bit along its circumference, then the value of 7r could easily be reduced to 3. [Exercise: Calculate the angle subtended at the centre of the circle by the

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“straight bit”.] Another friend of mine, Paddy Barry, told me that he once received a letter from a crank who claimed that he could find precisely the length of any curve from a secret new formula. Doubting he could do this, Paddy Barry asked to see his formula just in case by some chance in a billion the fellow had hit on something

significant. When the letter duly arrived, the first line of the proof read: “Without loss of generality, let the curve be a circle.”

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Parent to ten-year old having trouble with mathematics:“Look at it this way. If you had six pocket calculators and I took away four, how many pocket calculators would you have left?” One numerical analyst to another: “Does 315273063149237130832.31295410028397412 look right to you?” An eminent statistician with a good head for figures has calculated the collective measurements of the centrefolds of Playboy Magazine since its inception. The result was a supergirl weighing 17 tons with a chest measurement of nearly half

a mile.

Have you heard about the element which refused to belong to any set that would lower its standards so much as to accept it as a member? [The so-called Groucho element.]

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You know those books that give answers to the odd numbered problems only? One sadistically minded number theorist had the following exercises in his book:

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1* Show that the equation x3 + y3 = z3 has no solution for natural numbers x , y and z. 2** By induction, show that the equation xn + yn = zn has no solution for natural numbers x , y and z for n > 3. The most useful invention in mathematics is Skinner’s variable constant A. It is defined to be that number which when added to, subtracted from , multiplied by or divided into the number you’ve got, gives you the number you want.

Some students believe that a statistical correlation and causality are almost synonymous. It may be useful to remind them that it was once shown that there was a strong correlation between the number of bicycles sold in Glasgow and the number of illegitimate births in Birmingham. Have you heard about the mathematics professor who dreamed that he was ad-

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A company manufacturing hair restorer once received the following testimonial from a grateful customer: Dear Sirs, Before using your hair restorer I had three bald patches. Now I have only one. [Is this a joke about connected sets?]

There is of course the well-known proof by induction that all men are bald. We phrase the theorem as follows: If n is a natural number, then a man with exactly n hairs on his head is bald. Certainly the theorem is true for n = 1 because nobody will deny that a man with only one hair is bald. Assume true for k . Then a man with k + 1 hairs on his head is also bald, since the addition of one hair to a bald man’s head does not change his state of baldness. It follows by mathematical induction that a man with exactly n hairs is bald, for each n. Therefore all men are bald - the bald truth, one might say. A policeman one evening came upon a drunk who was on his knees crawling round and round a huge tree trunk which he was feeling with his hands. “Look sir”, said the policeman, “why don’t you just go home?” “I’d love to”, said the drunk, but I’m completely surrounded by this tree." [We may laugh at this, in fact I hope we do, but there are profound topological and spatial considerations involved. How can you find out if you are on the inside or the outside of something, say a sphere or space? What information does convexity give you?] An Irish mathematician has just made a major breakthrough on the dual of the four-colour theorem. In this problem about plane maps, any two countries having a common boundary must have the same colour. The Irishman has proved that seven colours are sufficient to colour any map no matter how complicated, and he conjectures that this is the best possible result.

From the proceedings of a conference on logic: Before we put the motion ‘that the motion be now put’, should we not first put the motion ‘that the motion “that the motion be now put” be now put?’ A great problem in applying mathematics is the question of how valid is a mathematical solution to a question based on a model, when we try to apply it to a real life situation. For example: If a young man can pick a pound of strawberries in half an hour and a young lady can pick a pound in forty minutes, how many pounds of strawberries will they pick if they are left alone in a strawberry field for an hour? Teacher: If there are ten crows sitting on a wall and I shoot one, how many will be_ _ left? T

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Teacher: You don ’t know much about arithmetic, Johnny. Johnny: And you don’t know much about crows, teacher.

“It’s called ‘ R0 green bottles hanging on the wall ’,” smiled the mathematician. [A cantor perhaps?]

Teacher: If Tom gave you three apples and Bill gave you two apples, how many apples would you have then? Mary: Seven apples, teacher. Teacher: Wrong, Mary, 3 + 2 = 5. Mary: I know that , teacher, but I have two apples already. [This is a joke about boundary conditions.]

A student, unable to finish a mathematical assignment on time, explained to his professor that he hadn ’t been feeling very well. “ Young man ,” retorted the professor,“most of the great work in mathematics was done by people who were not feeling very well.”

A recent report from the Association for the Promotion of Statistics states that this year meaningless statistics were up 4.29% on last year’s figures. An engineer, a physicist and a mathematician were marooned on a desert island . The only food they had was a can of beans and they had no means of opening it. The engineer wandered off looking for a rock with which to break the can. The physicist conducted various experiments by rolling the can down inclined planes. By calculating moments of inertia he was able to estimate the number of beans in the can , and he concluded that there wouldn’t be enough for a square meal for all three of them . The mathematician said , “Suppose we have a tin -opener.”

Many students fall into the trap of assuming that because a result is true for a large number of values of n, then it is true for all n. For each n, here is a theorem that is true in n cases but not true in general: All integers are less than n + 1.

Top of the best-selling books among statisticians this month is one entitled BOOK OF A MILLION RANDOM NUMBERS (listed in increasing order)

Jesus Christ could never have become a professor of mathematics. All are agreed he was a wonderful teacher, but he never published anything. Those who believe that numbers have no meaning in the abstract , may take consolation from the following extract from a trade journal : “We apologise profusely to all our customers who received as the result of an unfortunate computer error, the chest measurements of members of the Female Wrestlers Association instead of the amounts of soya beans sold to overseas countries”.

A mathematician was about to be executed by firing squad and was asked if he had any last request. “ I should like to sing a song,” said the mathematician. “Certainly”, said the chief executioner," which sone do vou wish to sine?”

Three boys at a juvenile court in England admitted to stealing £13.71 from a local shop. However, they threw away £1.71 because they couldn’t handle the division of the original sum.

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Have you heard about the fellow who was reputed to be cleverer than Einstein? Well, it was claimed that only four people understood Einstein’s theories, but nobody understood this fellow’s theories.

A mathematician once organised a raffle and the tickets cost £100 each. The prizemoney however was advertised as infinite, and he was inundated with applications for tickets. The lucky winner got £1 the first year, £5 the second year, £ 5 the third year. . .

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A mathematician was given a test in which he had to produce steam starting with a block of ice which was stored in a refrigerator. He successfully described in great detail all the steps involved in the procedure such as thawing the ice and boiling the water. Next he was asked to produce steam starting with the contents of a small pond. He replied , “Put a bucket of water from the pond into the refrigerator and apply the result of the previous problem.”

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A physicist , a statistician and a mathematician were travelling by train together in a strange country. The physicist looked out the window and saw what looked like a black sheep. “All the sheep in this country are black ,” he declared. “Come, come,” said the statistician, “you mustn’t jump to conclusions. All you can say is that some of the sheep in this country are black.” “Perhaps what you two are getting at,” said the mathematician, “is that in this country there exists at least one sheep which is black on at least one side.”

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A friend of mine received a note from his bank manager telling him that the interest rate on his loan had been raised to 12 per cent per anum. He replied that he would prefer to continue paying interest through the nose as previously if the bank had no objection

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A well known mathematician once visited a small college to deliver a lecture. The department was so obviously run down and money was in such short supply that he announced that he would waive his fee and look after his own travelling and flcrnmmnHatinn

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After the lecture the chairman thanked him profusely for his kind gesture and told him that the money involved was being put into a special fund. "What is this fund for?” asked the mathematician politely. “So that we can have better speakers in future,” said the chairman. i

The Jewish influence on both humour and mathematics is so strong that not surprisingly many mathematical jokes have a Jewish flavour . Here are a couple : Little Ikey was doing his homework but was having some problems with arithmetic. “What is 3 + 2, dad?”, he asked his father. His father beamed and took the little lad on his knee. ‘Tell me, son,” he asked,“are you buying or selling?” Jacob’s teacher asked him to calculate the interest on £6,317 for three years at 4 per cent per annum , but Jacob just sat there with a blank page in front of him . “What ’s the matter, Jacob, can ’t you do the question?” asked the teacher. “Of course I can,” said Jacob, “but who is interested in 4 per cent?” ••

A huge bridge collapsed, and the consultant engineer who designed it was arrested and charged with professional incompetence. At his trial he blamed everything and everybody except himself. He blamed the materials, he blamed the workmen and he blamed the supervisors. He even blamed the mathematicians who had produced the tables on which he had based his calculations. He said they were so ignorant that they spelled cos with an ‘h’. An hotel porter was asked by a guest what was the average tip he got for his services. ‘Ten pounds,” he replied. As the guest handed over a ten-pound note, the porter remarked, “You ’re the first person who’s been up to average since I started this job”. A physicist once proved that no odd integer could have a non-trivial factor. He cited 1, 3, 5 and 7. He had some trouble with 9, but on proceeding, he discovered that the pattern held for 11 and 13. Finally, he put 9 down to experimental error.

There is an old chestnut about two cars starting at the same time in opposite directions from the ends of a hundred-mile long road. Each car travels towards the other at a speed of forty miles per hour. At the start, a fly (of incredible prowess) leaves one of the cars and flies over and back between the cars until they meet, at a speed of sixty miles per hour. The problem is to find the length of the journey covered by the fly until the cars meet. The neat and simple solution is that the cars take 75 minutes to meet and so the

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fly, travelling at 60 miles an hour, travels 75 miles in that time. When this problem was put to the famous mathematician John von Neumann, he gave the correct answer almost immediately. “That ’s impressive,” said the questioner. “Most people try to do it by adding up an infinite series.” “That’s how I did it,” said von Neumann indignantly.

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De Moivre is one of the few mathematicians who died by an arithmetic progres sion. During his final illness he decided that he needed to sleep for a quarter of an hour longer each day than the previous day. On the day after he slept for twenty-three hours and forty -five minutes, he slept for twenty-four hours and then passed away peacefully in his sleep. [Exercise: Die according to a geometric progression.] The world ’s greatest and most powerful computer was constructed, so mathematicians decided to test it out by seeing if it could make any impression on some classical unsolved problems. They decided on Fermat’s last theorem, namely that xn + yn = zn has no solutions over the natural numbers for n > 3. For days, they fed it with every known piece of information, conjecture and partial result and at last they set it to work. After a few minutes it printed out: “I have a wonderful proof of this result, but my memory is too small to store it.” [ Is this what is meant by marginal progress?] The late John Rose, a wonderful mathematician who died so young, told me the following story that happened to him while he was teaching mathematics to an adult education class. There was one very keen middle-aged man in the class who was never late and never missed a lecture. He always sat in the front row and took copious notes. One evening Rose asked him how he was getting on and if he had any questions he wanted to ask about the course. “I’m enjoying it very much,” said the man, “and I’m really delighted to be getting a second chance of coming to grips with mathematics. Actually, I do have a question that is puzzling me a lot. What is the difference between a mapping and a lemma?” Since I am mainly interested in group theory, I was once amused to find that a pair of bedroom slippers I had bought bore the brand name COSET. However, there is a disturbing implication. Since I have a right coset which is not a left coset, does this mean I’m not normal? When God created all the animals he told them to go forth and multiply. A year later he visited the Garden of Eden to see how things were progressing and found that all the creatures had multiplied except the snakes. “What’s the matter?” he asked them. “We can ’t multiply because we’re adders,” said the snakes. So God chopped down some trees and made the snakes some log tables.

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The difference in outlook that exists between mathema tics and the arts is well illustrated by the correspondence between Charles Babbage and Alfred Lord Tennyson, the poet. Tennyson wrote a poem called “The Vision of Sin”, which contained the lines

Every minute dies a man, Every minute one is bom. Babbage rightly pointed out that if this were true then the population of the world would remain constant, whereas it was well known that the population of the world was increasing. He went on to suggest that Tennyson amend the lines to read Every minute dies a man, And one and a sixth is bom, adding that although the exact figure was closer to 1.167 men per minute he would not put such impossible strain on the metre of the poem. Not suiprisingly, Tennyson did not accede to Babbage ’s request, but in subsequent editions he did change the word “minute” to “moment”. There once was a mechanical engineer who had a great interest in mathematics. Year after year he religiously attended the seminars organised by the mathematics department of the college where he taught. Sadly, however, he found that he understood less and less of what was being said, and finally the talks became so technical that he stopped attending altogether. Then one day he received a notice that the mathem atics department was having a guest speaker the title of whose talk was, On the Mathematical Theory of “ Gears”. With eager anticipation and poised pen he took his place in the seminar room. The speaker began his talk with the words, “The theory of gears with a real number of teeth has long been understood.” Laplace was once asked who was the greatest mathematician in Germany. He replied “Pfaff ’. “ But what about Gauss?” asked the questioner. “Gauss,” said Laplace, “is the greatest mathematician in the world.” Teacher: Why are you scratching your head all the time? Student: I’ve got arithmetics in my hair. Teacher: Why do you call them arithmetics? Student: They add to my misery, they subtract from my pleasure , they divide my attention and they multiply like mad . A learned rabbi was known far and wide for the wisdom of his advice and many people came to him to settle their disputes. Once a man came and put his side of a dispute to him, and the rabbi said, “You are right, my son. ” Next day, the opponent in the case came and put his side of the stoiy to the rabbi , who said , “You are right, my son.” When the first man heard of this he rushed furiously to

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the rabbi and said , “ Yesterday you said I was right, today you said my opponent was right. You are right , my son,” said the rabbi.

An airline jet was out of control over the city of Warsaw and the pilot tried every manoeuvre in the book to stabilise it. Every device failed and the plane wobbled and vibrated more and more. Finally, the pilot had a brainwave: he asked all the passengers to move to one side of the aircraft. He had remembered that if all the poles are in the left -hand plane, you have stability.

The following exercise was found in a mathematics textbook published in 1820. If an archbishop can pray a soul out of purgatory in an hour, a bishop in three hours, a priest in five hours and a friar in seven, how long would it take them to pray out another sinner twice as wicked , all praying together? The answer given is one hour, 11 minutes and 35 seconds. [ Hint: Use cardinal numbers.]

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Mathematical Professor: If there are five continents and the Atlantic Ocean is four miles deep, and the distance from the North Pole to the South Pole is eight thousand miles, what age am I? Student: Forty-six, sir. Professor: That ’s incredible, how did you work it out? Student: Well, I have a brother who is twenty three , and he is only half crazy.

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A textbook on Arithmetic, published in Cork in 1831, has in its preface the following delightful disclaimer: As the hours for writing this work were so limited, and confined to the night, the Author hopes that any errors will be treated partially.

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Recently, an international competition was held to discover the world ’s most useless number. The winner was 1.8015446 x 1012, which claims to be the speed of light in furlongs per fortnight.

At a murder trial a famous applied mathematician was called to give evidence for the defence by saying that because of the rate of heat loss of the body, a certain murder could not have taken place at a certain time. “ What are your qualifications as an expert witness in this case? ” asked the judge. “I am the world ’s greatest authority on differential equations,” was the reply. “Isn’t that a rather pompous claim to make?” asked the judge. “ No," said the applied mathematician, “you must remember I’m on oath.”

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I heard the following at the British Mathematical Colloquium , spoken I think by that droll mathematician Martin Dunwoody: “We frequently find , well, it would be more correct to say we sometimes find, well , in fact, there is just one known example.. . ”

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The world ’s most powerful computer was fed with extracts from the writings of all the world ’s greatest philosophers in an attempt to discover the meaning of human life. The computer printed out: THE PURPOSE OF HUMAN EXISTENCE IS TO BUILD COMPUTERS It has been said that a mathematics professor is someone who learns more and more about less and less until finally he knows everything about nothing. A mathematics student, on the other hand, is someone who learns less and less about more and more until finally he knows nothing about anything. A little boy told his mother that he wasn ’t going to school any more because his teacher told lies. “ What do you mean?” asked his mother. “Well,” explained the little fellow, “yesterday he told us that two and two made four, and today he said it was three and one.” A newly-appointed schools’ inspector of mathematics was visiting a class in the East End of London . He asked the class to call out a two-digit number, and when the number 94 was called out he wrote 49 on the blackboard. Then he asked for another such number, and when 27 was called he wrote 72 on the blackboard . When he called for a third number a voice from the back shouted , 55. “ Now let’s see you muck about with that one.” There was once a statistician who travelled regularly by air between London and New York. He always carried a bomb in his suitcase because he figured that the probability of two people carrying bombs on the same flight was absolutely infinitesimal. One of my students once described my course as follows: This was a very comprehensive year of mathematics. Anything not actually mentioned in the course was very well covered in the final examination.

There are very few positive integers which can claim to have a story all to itself . Such a number, however, is 1729. As Ramanujan, the great Indian mathematical genius, lay dying, he was visited by his friend and mentor G.H. Hardy. To make conversation, Hardy mentioned the number of the taxi he had arrived in, 1729, and remarked that it seemed a very uninteresting number. Ramanujan is reputed to have raised himself from his deathbed and said feebly, “On the contrary, my dear Hardy. It is the smallest number expressible as the sum of two cubes in two different ways.” [Exercise: What are the corresponding numbers for squares, fourth powers, fifth powers etc.? How many solutions has the Diophantine equation an bn cn + = + dn for n > 4?] Three Irishmen were being given an intelligence test to decide who should get a job on a building site. Each was asked , “What is 2 2? ” +

The first answered “37” and was ruled out. The second answered “Wednesday” and he was ruled out also. The third answered “4”. “That’s terrific,” said the foreman. “How did you work it out?” “It was easy,” said the Irishman. “I just subtracted 37 from Wednesday”. [ Isn ’t it ironic that examiners in mathematics are allowed to give marks for an incorrect answer using the right method , but not for a correct answer using the wrong method ?]

The great Kronecker was once asked how he would go about advising a research student to choose a topic for mathematical research. He replied that he would sooner advise one of his students as to which wife he should marry.

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At a conference an eager young research student cornered a venerable old mathematician whose identity was unknown to him and asked him if he had read the literature in a particular area of mathematics. “Read it?” thundered the venerable one. “I’ve written practically all of it.”

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Douglas Munn told me the following lovely story about his visit to the International Congress of Mathematicians held in Stockholm in 1958. He was an eager young research student studying algebra at the time, so he sat up near the front of the hall at the first algebra session with pencil poised and a brand new notebook . The first speaker opened his talk with the words, “We shall assume our topos is

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The mathematician J.E. Littlewood once attended a conference on the Continent and was astonished to be greeted thus by a French delegate: “So there really is a Littlewood! I thought it was just a pen-name under which G.H. Hardy wrote his inferior papers.” A mathematician was out walking in the woods with his girlfriend. She picked a daisy and began to play “He loves me, he loves me not,” pulling off the petals one by one. “Really, my dear,” he said to her, “you should adopt a more precise approach. Count the number of petals on the flower. If the number is odd then he loves you, and if it is even he loves you not. It’s as simple as that."

In 1941 an Oxford mathematician was asked why he wasn’t in France fighting to preserve civilization. “I,” he replied to his questioner, “am the civilization they are fighting to preserve.”

A statistician was about to undergo a serious operation and asked the surgeon what his chances of survival were. “ Your chances are excellent,” said the surgeon. “ Nine people out of ten die from this operation , and the last nine patients I’ve operated on have died.”

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When Albert Einstein first arrived in the United States, he was invited to speak at Princeton University. The large hall was packed to capacity with people who had come out of curiosity to hear the world-famous scientist. Einstein was amazed at the huge crowd and remarked to the chairman , “I never realised that so many Americans were interested in tensor analysis.” The mathematical reputation of the University of Cambridge is the envy of the rest of the world. This reputation is said to be maintained by the method in which (it is claimed) honours in mathematics are awarded at the degree examination. The candidates’ scripts are taken to the top of a large staircase and allowed to slide down the stairs. Those reaching the bottom are awarded firsts, those reaching half way are given seconds and the rest thirds. Clearly, this system is just as fair as those used in other universities. A very precise mathematician was in the habit of saying, “Consider the real polynomial ax4 + bz3 cx 2 + dx + e, where e need not be the base of natural logarithms.” A friend of mine, Martin Newell, once told me that he delivered a seminar on group theory in which he proved several theorems about p-groups. A member of the audience felt that the theorems in question were true for all groups and asked him to show where the “p-ness” came in.

A famous mathematician lived in retirement in a country village in England during the Second World War. As there was a shortage of teachers, he was asked if he would take classes for a few weeks in the local school to teach some elementary probability theory. The usual method of teaching these students was to roll dice with them for several weeks and let them win lots of money in order to motivate them. The retired professor opened his course by turning his back to the class and writing on the blackboard :

Let B be a Borel field. Einstein ’s wife was once being shown around a huge astronomical observatory. She pointed to one very large telescope and asked what it was used for. “ We use that to discover the structure of the universe,” said the director. “Really?” she replied. “My husband uses the backs of old envelopes to do that.” There has been a great deal of discussion over the last few hundred years as to what is the essential difference between a pure mathematician and an applied mathematician. The following is perhaps the closest that anyone has come to pinpointing the difference in outlook. There is a square room of side twenty feet with a pure mathematician in one comer and an applied mathematician in the opposite comer. In a third comer is a voluptuous girl. The mathematicians are allowed to approach the girl in

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bounces along the sides of the square, the first bounce a maximum of ten feet and every subsequent bounce a maximum of half the previous bounce. The pure mathematician, well versed in limits, quickly calculates that no matter how many bounces he takes he can never reach the girl , so he doesn’t even begin to bounce. The applied mathematician sets off at once because he realises that after five or six bounces he will be close enough for all practical purposes. [Should applied mathematics be called “impure mathematics?”] When I told this story recently in class, one of my female students said , “Why should the girl stay in the comer waiting for the idiot bouncers to arrive?” A fair point.


« < , ? Oh ” said the mathematician. “don’t they communicate?”

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A former student of mine, David Flannery, was teaching a class about binary operations. He went through the usual examples of addition and multiplication and then gave the general definition of A B “Now,” he asked his students, “where have* we seen something like that before?” An enthusiastic hand shot up. “M*A *S*H”, said the student. A friend of mine asked in i a university library for McCoy’s book “Rings and Ideals”.

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“What category is that in?” asked the assistant in all seriousness, “ romantic fiction?” was asked in an examination to write an essay on Einstein’s claim that

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A student space is curved . He began as follows: We all know what ‘space’ is. We all know what ‘curved ’ means. The whole claim depends on what Einstein meant by ‘is’. . The famous inventor Thomas Edison had a very low opinion of mathematicians and sum a out work them to ask “They make me tired ,” he used to say. “You y’s; s they take a piece of paper, cover it with rows of a ’s and 6’s and x’ and around decorate these with a lot of little numbers; scatter a mess of fly specks them , and then give you an answer that’s all wrong.” day Edison once hired a mathematics graduate to help him in his laboratory. One to he gave the young man a new light bulb he was working on and asked him to ns , calculatio some did rule slide his took calculate its volume. The young man to calculus , used then and approximate the equation of a vertical section outline find the volume of revolution . After an hour of checking he presented his results to Edison. “ You ’ re at least 30 % out ,” said the great man. “ I ’ve just calculated the volume by taking the top off and filling the bulb with water.” The following recommendation was made with regard to a difficult undergraduate who had applied for a place on a postgraduate mathematical course: “We feel this student would be much happier in a larger or smaller college.”

An old carpenter was using a wooden folding ruler from which all the numbers had long since worn off. His boss asked him what possible use such an instrument could be to him. He replied that he had such a good memory that he could remember exactly where all the markings were. A famous story is told about the theoretical physicists Hoyle and Bondi. It is usual in astrophysics to work with units that make all fundamental constants equal to unity. Then a conversion factor is applied to get the answer in standard for units. Bondi calculated some important number in physics and asked Hoyle the conversion factor. “It ’s 1060,” said Hoyle. Bondi replied , “Do I multiply or divide?” A mathematician was being shown around a prehistoric museum by the curator. When they came to the dinosaur section, the curator said, “This is the oldest dinosaur in our display. It’s twenty million and seven years old.” “How can you be so definite about its age?” asked the mathematician. “Well ”, said the curator, “it was twenty million years old when I started work here seven years ago.”

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A professorship in mathematics at the University of Oxford sometimes involved minimal lecturing duties. One appointee was in the habit of delivering a lecture on the subject once every three years. When it was suggested that he might increase his output to once per year he replied, “I don ’t think so, one can overdo that kind of thing, you know.” A friend of mine once gave what he regarded a superb lecture on limits. His definition was clear, his examples constructed to bring out the of the situation, and his overall performance designed to stimulate the subtleties students’ interest. He then asked if there were any questions and was shattered to have a student ask, “What is the value of those numbers e and £?” I once sent the following letter to the editor of a well known mathematical journal: Dear Sir, I would like to tell you about an amusing mistake I found in an examination paper I was correcting. One of the questions was, “ What is meant by saying that the limit of f ( x ) as x approaches a is equal to / ?” A student

answered: The limit of f ( x ) as x a is equal to / if given any number 6 > 0 there exists a number e depending on 6, such that | (i) / | 6 for all < / 0

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There is a nice story, possibly apocryphal , of a Scottish mathematician who wrote a paper on number theory in Scots Gaelic, using the Gaelic version of his name. The paper was sent to him to be refereed in the English version of his name, as he was the only expert in number theory fluent in Scots Gaelic. He reported that the paper was an excellent one ( which it undoubtedly was) and he strongly recommended publication. Naturally, the paper was accepted.

Has it come to this? Recently I heard one young fellow say to another, “My calculator is smarter than your calculator.”

do” was found on the desk of G.H. It is said that the following “List of things to Hardy. 1. Prove the Riemann hypothesis. of God that will convince the man 2. Find an argument for the non-existence in the street. 3. Murder Mussolini. who first discovered the connection Some time ago, Claud Shannon , the man theory, invented a new computer. It is called between boolean algebra and circuit only. Now that ’s progress. THROBAC and it calculates in Roman numerals he was asked by a friend if it was After Bertrand Russell’s first child was bom a boy or a girl. He replied, “ Yes.” mathematician. This one concerns the Yet another story about an absentminded on his door and went off to give Cambridge professor who put an “OUT” notice , which of course he had forgotten , a lecture. When he returned to get his notes sadly away. he saw the “OUT” notice and turned dining with him. After dinner An old clergyman once had two mathematicians sleep in the middle of their off to they talked shop and the clergyman dropped your pardon, Mr. X. I fear I beg , I conversation. Waking abruptly, he said “ Mr. X replied, “ Not at all. It dropped off to sleep while you were speaking.” asleep.” was Mr. Y who was speaking when you fell is transcendental. To show It is not known whether or not Euler’s constant 7 resign his chair in favour to how hard the problem is, G.H. Hardy once offered . irrational was it of anyone who could even prove that journal, received some very The great Hadamard , as editor of a mathematical , so he invited him to dinner. The good papers from an author unknown to him his control , he would not accept man replied that owing to circumstances beyond did, and _ . he invited Hadamard to visit him. This Hadamard the invitation but lunatic criminal a confined to found to his great surprise that the author was the murder of several of his asylum. Apparently he was quite sane except for very good mathematician. aunts. His name was Bloch and he was a

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the humour of pure math Up to this point this book has been concerned with to a maiden aunt or even ematics, so pure in fact that it could safely be given introduce a little impure a retired topologist. I beg the reader’s indulgence to secondly because it is ab , mathematics, firstly because it is applied mathematics innocent and shows what solutely genuine and thirdly because it is all perfectly case of honi soit qui mal y time can do to mathematical terminology. Let it be a

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pense It concerns a textbook entitled “ A Treatise on the Theory of Screws” published in 1900 by the Cambridge University Press. It was written , appropriately enough, by Sir Robert Stawell Ball LL.D. F.R.S., Lowndean Professor of Astronomy and Geometry in the University of Cambridge and formerly Astronomer Royal of Ireland. The book contains 540 pages and 434 sections, and the titles of some of these sections make present day sex manuals seem very tame indeed. Incidentally, there is a consistent misprint throughout , where “ wrenches” is used when “ wenches” is clearly intended . Here is a selection of the titles of some of the sections , taken from the table of contents. 2. Definition of the word screw 6. Instantaneous screws 7. Definition of the word wrench 20. Reciprocal screws 25. Screw reciprocal to five screws 29. Intensities of the components 30. The intensity of the resultant 31. Co-reciprocal screws 38. The pitch 45. Indeterminate screws 46. A screw at infinity (never the twain shall meet) 80. Impulsive screws 81. Conjugate screws of inertia (does your marriage suffer from this?) 90. Twist velocity acquired by an impulsive wrench 92. Formula for a free body 104. Harmonic screws (to music?) 106. Discussion of the results 114. Small oscillations 134. Body with two degrees of freedom 150. A screw that will acquire a given twist velocity under a given impulse 186. The ellipsoids of inertia 202. Imaginary screws 206. Screws at right angles 247. Double screws 248. The seven pairs 298. Three pairs of correspondents 323. Two rigid bodies 354. Reciprocal screw-chains 368. The restraining wrench chain 374. Permanent screw-chains 378. Restraining wrench for a free rigid body

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386. The permanent screw The final chapter concerns screws in non-euclidean space, and caters for all tastes, as is evidenced by 428. Two homonymous vectors compound into one homonymous vector

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A story, seemingly true, is told that Erdos was visiting a Texas University and over coffee noticed a problem on the blackboard. It concerned functional analysis about which he knew only the bare minimum. It seems that two local analysts had just come up with a 30-page solution to the problem of which they were very proud. Erdos looked at die problem and asked to have it explained to him. He then asked a few more pertinent questions and within a few minutes solved the problem in a few lines.

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Thomas Hobbes, the philosopher, must have been one of the few people to have approached geometry from the correct angle. This account is given by Hobbes’ friend John Aubrey. He was 40 yeares old before he looked on geometry ; which happened acciden tally. Being in a gentleman' s library Euclid' s Elements lay open , and ' twos the 47 El . libri I . He read the proposition. ‘ By G ,’ sayd he , ( He would now and s a y d he , ‘ this is impossible!’ So he then sweare , by way of emphasis ) ‘ By G reads the demonstration of it , which referred him back to such a proposition; which proposition he read . That referred him back to another , which he also read . Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.

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fences they use to keep the cattle under control.” [ Yes, this story does belong in this book .]


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still get different rooms but room 1 is now left free. realises that f ( n ) = When at 9.30 p.m. 100,000 additional guests arrive, the clerk full hotel and already an into guests n + 100 , 000 will squeeze 100,000 additional manner. same the in in fact any finite number of guests can be accommodated to which hotel the infinite At 10 p.m . an infinite bus arrives which pulls up outside everybody fast , now asks it looks remarkably similar. The clerk , who is learning the one-to-one function using again once to double his room number and move vacant which neatly numbers / ( n) = 2n. This leaves the infinite. corridor of odd accommodates the infinite busload are accommodated using At 10.30 p.m•» 729,613 infinite buses arrive and these be used to accommothe function f ( n ) = 729 , 613 n. Clearly this technique can date any finite number of infinite buses . Again the existing guests At 11 p.m . an infinite number of infinite buses arrives. vacant and the are asked to double their room numbers leaving the odd corridor scat number of new arrivals are indexed by N x N , their unique combination and bus number.

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) ( 1 ,2), (2,2), (2, 1), These people are now given vacant rooms in sequence ( 1 , 1 , (3 , 1 ), ( 3,2), . .. as per the arrows in the diagram . suddenly receives The clerk is now on the point of a nervous breakdown when he to supply a daily him a phone call from the company ’s financial controller asking hotel bill ( 1 ) or account of those rooms where the guest is up to date with his all possible such not up to date (0). The clerk , for convenience , decides to list ’ process shows diagonal s Cantor sequences of ones and zeros. Need we go on? that this quest is doomed to failure. morning Suddenly the cleric sees a good way of solving all his problems. Next clean the rooms. at 10 a.m . he calls in an infinite number of chambermaids to

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In 1972, Paul Erdos started putting the letters PGOM (Poor Great Old Man) af ter his name. At 60 he became PGOMLD (...Living Dead ) and at 65 he became PGOMLDAD (...Archaelogical Discovery). At 70 it grew to PGOMLDADLD (...Legally Dead ) and in 1988 it became PGOMLDADLDCD (...Counts Dead ). When his friend Paul Turan died in 1976 Erdos had an image of the Supreme Fascist (God) weighing all his papers on a balance every time one of his collaborators died. On one side He would place the papers Erdos had co-authored with the living and on the other side those co-authored with the dead. Erdos jokingly predicts that when the dead side tips the balance he too will die.

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Ian Stewart once proposed that mathematicians should sell their ideas and copyright their theorems. This was to counteract the huge salaries paid to computer scientists who sell their software to the public. However, on further consideration he decided against the proposal , imagining for example the damage that might have been caused to mathematics if the Isaac Newton Corporation (Fluxion Inc?) owned all rights to the calculus after a multi- billion legal action against G.W . Leibnitz Limited (geddit?).

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From L.E. Dickson ’s description of his honeymoon: My wife and I both agree it was a great success though I only got two papers written.

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The famous number theorist D.H. Lehmer once came up with a sequence of numbers which were quite mysterious. He looked them up in Sloane’s "Handbook of Integer Sequences” and found a reference to a journal article published in the 1930s. On looking it up he found it was a paper of his own! Moreover, he had come across the numbers in a totally different way and so was able to relate the two approaches to produce a much more interesting paper.

When lecturing about chaos a mathematical lecturer illustrated one point by saying that the earth’s weather system was so unpredictable that the fluttering of a butterfly’s wings in Siberia could tip the balance of weather in the United States from a hot sunny spell to a prolonged blizzard. When he asked the class for comments, one student suggested , “Let ’s kill all Russian butterflies.”

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for 100 marks, so each A mathematics teacher gave a test with 9 questions got all the questions right and question scored 11.1111.... marks. Young Smith so got 99.9999.... marks. the questions right, but I didn ’t “How is it,” he asked the teacher, “that I got all get full marks?” “Well ,” said the teacher, “that’s the limit.” morning and happened to remark to The owner of a factory arrived at work one London by train. the nightwatchman that he was about to go to , “ I had a terrible dream last alarm “ Don ’t go, sir,” said the nightwatchman in and you were killed .” night about your being on a train to London which crashed “ You ’re fired ,” said the owner. a simple logical deduction.] [To “get ” a story it is sometimes necessary to make

: Samuel Johnson’s definition of a network is noteworthy , with interstices between Anything reticulated or decussated , at equal distances the intersections. which contains all the Paul Erdos once claimed that God has a transfinite book theorems, including many best , neatest , and shortest proofs of all mathematical ian does something which still await solution in this world . When a mathematic a glimpse of a page given neat and clever what has happened is that he has been that Erdos can pay a proof in the book for an instant. The highest compliment is that it is “ straight out of the book.” , not for the first time , they I once tried to get this book on inter-library loan and replied “no known location .”

al department of A friend of mine once got a job as an assistant in the mathematic before Scottish University noted for its strictness. One day he was called

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astonished National Insurance contributor. The computer had been distracted by the static electricity from the nylon garments worn by the girl operators.

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Here is a piece of very valuable advice to any young mathematics lecturer starting his career: if you find you are getting more laughs than usual with partial differential equations, just make sure that your trousers are properly zipped up.

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I once submitted the following paper to a well known monthly mathematical publication.

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A PAPER IN WHICH THE TITLE IS LONGER THAN THE STATEMENT OF THE RESULT WHICH IS IN TURN LONGER THAN THE PROOF OF THE RESULT.

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Theorem If a real sequence is both arithmetic and geometric then it is a constant sequence .

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Proof Let the sequence be a a2 d? => d = 0. The paper was not published.

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John Conway has a good test to decide whether or not a paper you have written should be published . Just after you have finished typing a paper, the devil comes to you and offers you £1,000 (this figure would be inflation-linked ) for you to tear up the paper and forget all about it. If you even hesitate, the paper should not be published.

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W.D. MacMillan was a colourful Texan who worked in mechanics and potential theory. According to legend , he was without a college education when he went to Chicago as a mature man to sell his cattle. Having sold them , he went to a Chicago Observatory to see the stars through a big telescope. He was so fascinated by what he saw that he stayed on and took a series of degrees with highest distinction and became a member of the faculty at the University of Chicago. During lectures, he used to say “people who ask ‘ what is mathematics good for?’, they make me tired. Like when you show a man the Grand Canyon for the first time. Then he turns to you and asks ‘What’s it good for?’ What would you do? Why, you would kick him off the cliff ’. Then for emphasis he would kick a chair halfway across the lecture room.

Modem version of the pigeonhole principle for today’s students. If m male pigeons make love with n female pigeons, and m > n then at least two male pigeons must make love with the same female pigeon.

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theory involving a parameter n A graduate student had a problem in number , decided to compute the value of and , unable to make any theoretical progress n using the most powerful computers the function for the first five values of available at the time. happened to be visiting the area so the Next day the great John von Neumann student mentioned the problem to him . like,” said von Neumann, and immedi“Let’s see what the numerical results are ately rattled off the results for n 1, 2, 3.. . Telling himself that he now had a “ Now let ’s look at n 4,” said the maestro of the century, the student picked chance to impress one of the greatest minds a minute gave the answer as up a pen, made as if to calculate, and after about 37.314. at him coldly. “ How about n = 5?” “That’s correct,” said von Neumann , looking ”. After about two minutes, the student said , “792.913 ns and said, “ That’s calculatio his Thirty seconds later von Neumann finished quickly?” absolutely correct how the hell are you doing it so a reward of $10 to any student An American mathematics professor offered proof - read textbook. One student who could find a misprint in his meticulously headline on a certain chapter scooped the jackpot by spotting that the running . had a misprint which netted him $10 per page is that one is expected to fill One of the hazards of being a mathematical lecturer to student job applications. in official references and questionnaires in relation extract from such a form: The following ( I am reliably informed) is an ? Q. What is the applicant’s standard of attainment A. Below average ability. employment as a government Q. Is the applicant , in your opinion , suitable for statistician? A. Yes. might like to ponder on the fol Those who think mathematics is complicated . lowing genuine extracts from official publications e as they would normally 1. Part time employees will be paid for such attendanc paid or given any additional have given but for the holiday. They will not be had there been no holiday in lieu if they would not have been in attendance holiday on the day in question. a time to work in a room 2. The number of persons habitually employed at by that number which shall not be such that the quotient derived by dividing of the room is expresses in square feet the area of the surface of the floor d number mentione first the less than 40 or the quotient derived by dividing 400. than less is which expresses in cubic feet the capacity of the room the amount by which any sum 3. The Authority shall pay to the Contractor

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payable on the basis of the prices finally fixed exceeds any sum paid on the basis of the provisional prices and the Contractor shall pay to the authority the amount by which any sum paid on the basis of the provisional prices exceeds the sum payable on the basis of the prices finally fixed.

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Next time you consult a set of mathematical tables it might be well to remember that one famous compiler used to plant several deliberate errors in his tables so that he would be able to spot his own calculations when others reproduced them without permission. How about a calculator with the same facility? An Irish mathematical friend of mine, who for obvious reasons must remain nameless, uses the following device for importing spicy books into Ireland. He puts in the top of the parcel three books on tensor analysis and he reckons that the fourth book down has never been touched.

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Tartaglia, whose name is associated with the first solution of the general cubic equation, was so poor that he received very little education. At the age of 14 he was sent to a teacher to learn the alphabet, but the money ran out when he got to the letter K. [Maybe this motivated him in the study of cubics: if ax* + bx2 +cx+ d 0, 1 know what a , b , c and d are, but what is x?]. Undeterred, Tartaglia stole a copybook and learned to read and write, often using gravestones as slates for want of paper to write on.

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Who was the most precocious of all mathematicians? George Boole could count at the age of | l , and Gauss corrected an error in his father’s calculations at the age of 3. They must be serious contenders.

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There is danger in small numbers. In pre glasnost days there was a car race between an American car and a Russian car. The American car won the race easily, but Russian newspapers carried the following report: In a recent international car race, the Russian car came second while the Amer ican car finished next to last.

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A mathematician once submitted to a journal a paper entitled “Some algebraic identities involving large numbers.” It was returned with the comment, “There is no such thing as a large number.”

editorial policy It seems that the Journal of Number Theory has a longstanding methods to Fermat’s last not to referee or accept papers which apply elementary does not exist? It seems theorem. But can they prove that an elementary proof that the Journal of Logic has a longstanding. • • western civilization to the The history of mathematics concentrates too much on are well authenexclusion of eastern civilizations, especially the Chinese. There before Newton long calculus integral ticated accounts that the Chinese invented a Chinese mathematician and Leibnitz, for example. Now there are reports of in mathematics long bewho discovered many of the concepts commonplace , and he discovered such fore western mathematics did so. His name was Co , cohomology, , codimension , things as codomains, cosets, correlations covariants coherence, collinearity and concurrency. was describing of In his first lecture at the university, a new professor statistics coin from his a removed , he the laws of probability to his students. To illustrate around a few , spun pocket and tossed it in the air. It landed on the polished floor to rest - vertically on times and to prolonged applause from the students came its edge. at about 109 to 1.] [The odds against such an event have been reliably estimated who believed he was an The great Isaac Newton was pestered by a woman she came to him every ring gold a astrologer with magic powers. When she lost her he told her that day for a week hoping he would find it for her. To get rid of would find it at her if she walked a hundred paces down a certain laneway she feet . She did . . In a cage was a G.H. Hardy once visited a zoo in Sweden with Marcel Rietz sniffed at the lock, bear, and there was a gate with a lock on the cage. The bear away. hit it with his paw, growled a bit , turned around, and walked not carry them , does but “He is like Polya,” said Hardy. “He has excellent ideas out.” in years, so, since A professor of the history of mathematics was getting on on tape and have a his lectures were always the same, he decided to put them look technician play them to the class each day. One day he thought he would classroom. He saw to see how his scheme was working, so he peeped into the , with no human recorders his tape machine speaking to a hundred students’ tape present in the room. in Cambridge, When Olga Taussky accused Philip Hall of being the worst recluse Hall replied “ No, Turing is worse.” . Steven Krantz There are various degrees of truth of a piece of mathematics of the word relates that in Princeton in the 1950’s there were four definitions , then it was “obvious.” If something was obvious in the sense of Beckenback

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true and you could see it immediately. If something was obvious in the sense of Chevalley, then it was true and you could see it within a few weeks. If something was obvious in the sense of Bochner, then it was false and it would take you several weeks to see it. Finally, if something was true in the sense of Lcfschetz , then it was false and you could see it immediately. There was also the concept of “ true in the sense of Cardan.” This arose when a mathematician called Grauert could not find a counterexample within an hour. Do you know those mathematicians who write in the air with a finger when talking to you about mathematics? I have a friend who does just that but when he is finished calmly rubs it all out!

Mathematicians seem to have adopted the “Halmos Tombstone” symbol to show that a proof has ended . It lends weight to the opinion that a proof is the obituary of a dead piece of mathematics.

Diderot (1713-84), the French philosopher, was visiting the court of Catherine the Great at St. Petersburg, where the great Euler was a member of the Academy. Diderot , a devout atheist, began to convert the courtiers to his way of thinking, much to the disgust of Euler, a devout Christian. At Catherine’s request, Euler was asked to silence Diderot, mathematically if possible. Before the court , Euler confronted Diderot and said in a stentorian voice, “ Monsieur, (a + bn ) n x , / = done Dieu existe. Ripondez." Diderot, who knew no mathematics, is said to have retired amid confusion and laughter, and the next day he was granted permission to return to France. The following definitions could not be omitted from this book , and I was fortunate in finding what / trust is a plausible mathematical guise for them:

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COLLEGE LATTICE The President Leaps tall buildings in a single bound Is more powerful than a train Is faster than a speeding bullet Walks on water Gives policy to God The Head of Department Leaps short buildings in a single bound Is more powerful than a steam engine Is just as fast as a speeding bullet Walks on water if sea is calm

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Professor Leaps short buildings with a running start and favourable winds Is almost as powerful as a steam engine Is faster than a speeding arrow Walks on water in an indoor swimming pool Talks with God if special request is approved

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Another probably apocryphal Norbert Wiener story relates that to emphasise a point in statistical mechanics he said , “This event is as improbable as a bunch of monkeys having typed out the Encyclopaedia Britannica.” Then he thought for a moment and reflected, “But of course that has happened once already.”

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A final Wiener story. A student once asked Wiener to solve a problem for him , so Wiener thought for a few moments and wrote down the answer. Now what the student really wanted was an explanation of the solution method, so he politely asked Wiener if there was some other way of doing the problem. Wiener thought again, smiled and said, “ Yes, there is,” and wrote down the answer again. G.H. Hardy once told Bertrand Russell that if he could find a proof that Russell would die within five minutes he would of course be sorry to lose him, but the sorrow would be quite outweighed by pleasure in the proof. I once asked a famous old mathematician if he had ever taken a course on spherical trigonometry. “No,” he replied, “but I’ve given one.”

There was once a farmer who was worried that his cattle herd was not giving enough milk, so he asked the local university to investigate the problem. A research team was set up headed by a world famous applied mathematician. After six months and a huge research cost, the team produced a 200-page result of their findings. The opening sentence was, “Consider a spherical cow.”

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And now a final flourish of limericks , beginning with what is probably the best of all the mathematical ones , given to me by David Singmaster:

A question both deep and profound Is whether a circle is round. In a paper of Erdos Written in Kurdish A counterexample is found. As Descartes lay flat on his bed A fly buzzed about overhead. “I think, so I am, And that fly’s in a jam When he reaches the point (x, y, z).” If I were selected and given The keys of the kingdom of Heaven, I’d leave in the cold All that legion untold Who think TT S twenty two over seven.

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A student from Warwick once said , “I’ll take mathematics to bed. My girl isn ’t willing, But I still want some thrilling, So I’ll integrate quietly instead .”

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A lad of the brainier kind Had erogenous zones in his mind. He liked the sensations Of solving equations; Of course in the end he went blind. A student of nuclear fission Built a bomb with official permission, But the Earth disappeared In the bang: it is feared Through an error in simple addition.

4 HOWLERS AND BONERS

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“I’m fed up,” said a right-angled triangle, “And I can’t stand this cursed right angle. “And what is the use “ Of a hypotenuse “If it doesn ’t go jingle or jangle?”

Mathematics is a subject where mistakes are easily made, and the more elemen tary the standard of material the more hilarious the mistake. On this side of the

Atlantic we call outlandish mathematical mistakes “howlers,” while in America they tend to be called “boners.” Perhaps the best known is the “law of universal linearity” which is frequently used in assignments and examinations. It is often quoted to justify the following theorems: (i ) (ii ) (iii ) (iv) (v )

However, this situation can be turned to advantage by setting such questions as (a) For what numbers x and y is it true that l (x y ) 1 x 1 y? / + = / + / (b) Find matrices A and B such that det ( A B ) det A det B . + = + Howlers and boners are frequently seen as very funny by the better students because they know what the correct answer should be. Weaker students see nothing to laugh at , and perhaps this is part of their problem. Finally, a word of warning: it is dangerous to tell even the best howlers to some students, because sometimes it is the only part of the course they remember and you run the risk of having them faithfully and accurately given back to you in examination scripts.

Many howlers are probably apocryphal and manufactured by teachers, but the first few are authentic , because / myself have collected them over the years

to relieve the brain-damaging tedium of marking scripts. They are possibly a dreadful reflection of my own performance as a teacher , but this is not a time for false modesty: Question: Prove that

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/ 3 -\/2 is an irrational number.

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Q: Show that the function / : Z A: / (1) = 1, so f is one-to-one.

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- [2 = 0.318 = \/ n , which is irrational .

An oxygen is an eight -sided figure.

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= 2x - 1 is one-to-one.

Parallel lines do not meet though produced to eternity. A curve is the longest distance between two points.

Using Binomi ’s theorem . .. [ Binomi was probably an Italian Renaissance mathematician ].

An angle is a triangle with only two sides.

Using Trichotomy’s Law . . .

A polygon is a dead parrot.

[Trichotomy is undoubtedly an ancient Greek .]

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An average is a thing that hens lay their eggs on - for example, “My hens lay four eggs a week on average.”

This integral can be evaluated using Gamma ’s function. [Gamma is probably another Greek.]

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A circle is a round straight line with a hole in the middle. Q: Decide whether or not the groups H and K are isomorphic. A: H is isomorphic but K is not.

Q: Show that c > R 0 , the cardinal of the natural numbers. A: As c is the velocity of light , and nothing can travel faster than light, therefore c > Ro

If one angle of a triangle is more than 90 degrees, the triangle is obscene. A polygon with seven sides is called a hooligan.

Q: In the equation E = hu , inteipret each term. A : E is the energy, h is Planck ’s constant and v is the length of the plank.

The circle passing through the vertices of a triangle is called the circumcised circle.

A circle is the locust of all points equidistant from a given point.

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Most howlers however are perpetrated in elementary mathematics by confused young people. Here are some classics:

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Infinity is where things happen that don ’t.

Parallel lines never meet unless you bend one or both of them. Q: A man has x miles to travel. He goes a miles by train, b miles by boat and c miles on foot. How far has he left to travel?

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Algebraic symbols are used when you don ’t know what you are talking about. An axiom is something so visible that it is not necessary to see it.

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The following solution was submitted as a solution of the differential equation dy / dx = y / (sin x ):

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The circular points at infinity are called Isaac and Jacob.

A point has position but no magnetism.

One of my students attempted to answer a question concerning x, the mean of the numbers xi , x2 , . . . , x„, using the undoubted fact that x is the complex conjugate of x.

One of my favourite howlers is absolutely authentic; it happened as a result of a question I set on a first-year college examination. I asked the students to draw rough sketches of a number of curves, among them y \/4 - x2. The method = expected was that the student should plot a number of points and use elementary

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= / 4 - x2 one student y

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A proof that a lazy young dog is a sheet of graph paper: 1. A lazy young dog is a slow pup. 2. A slope up is an inclined plane. 3. An ink-lined plane is a sheet of graph paper

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Let / (a:) be a manic polynomial.

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I once corrected a geometry paper in an elementary examination. In one question the students were asked to identify a certain figure, which was in fact a parallelogram. I collected the answers given in 300 scripts. They included the following: A diamond , a kite, a rhombus, a rectangle, a square, a pentagon, a trapezian, a cyclic quadrilateral , a quadratic, “a parallelogram (if we look at it sideways).”

It is my belief that electronic calculators should not be placed in the hands of any person under the age of 21.1 once set a question to an elementary class involving a triangle \ AB\ = 8,|J3C| = 7 and lABC = 42°. The students were asked to find \ AC \ using the cosine formula. One student had the value of cos 42° correct to eight decimal places, but on the same line gave 82 = 84. An example of the failure of the law d [ x , y ) = d ( y , x ) is that it is longer from New Year’s Day to Christmas Day than it is from Christmas Day to New Year’s Day.

This howler is surely a topological one: Moths survive easily because they eat only holes.

The basic postulates of geometry are a pencil , a ruler, and a rubber. i

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A cyclic quadrilateral is an animal that has been taught to ride a bicycle. Q.E.D. means “quite easily done.”

The graph of y = x as drawn by a computer science student. The temperature in Spain is 28°C . The temperature in London is 14°C . Therefore it is twice as hot in Spain as it is in London.

This, sadly, is an extract from a script written recently by one of my own students ( where did / go wrong , where did 1 go wrong? ) : Question: A committee of three men and four women is to be chosen from seven men and eight women. In how many different ways can this be done? Solution: 7 + 8» 3 4i - 12

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(iii) This series is not an arithmetic progression therefore it is a geometric progression.

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l ways or 3 ways if the people are doubled. This committee can be formed in| The only possible explanation is that the problem in question was the second part of a problem whose unrelated first part asked for a proof of De Moivre’ s Theorem. Incidentally, this particular student did not pass the examination.

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a moral to this last one. When teaching APs and GPs one shows students examples of APs and GPs. How many of us show students an example of a sequence which is neither an AP or GP? Otherwise a student might very well conclude that any sequence is either AP or GP.] [There is of course

A square is a circle with four comers.

[Perhaps this is good topology?]

Pythagoras was the first person to breed a hypotenuse in captivity. An equation involving x7 is called a septic equation.

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Have you heard about the Irishman who sold a £10 parking ticket to an Englishman for £51 [o > b =>• a > bl ]

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Irish Preacher: The bible says you should give a tenth of your income to the church. I suggest that because of inflation you should give as much as a twentieth , [o > b > ?]

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Irishman who joined the 75th regiment of the army to be near his brother who was in the 76th regiment? [The ordering of N , which makes 75 and 76 adjacent, does not induce an ordering on the set of regiments indexed by IN .]

How do you recognise an Irishman’s word processor? The screen is covered in Tippex (Whiteout )

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An Irishman was travelling on the London Tube late one evening when he saw a notice saying DOGS MUST BE CARRIED ON THE TUBE “Where am I going to get a dog at this hour of the night?” he moaned. An Irishman won a round-the-world cruise in a competition. He refused to accept his prize because he said he had no way of getting back. [This seems to be a very deep joke Torsion is certainly involved, but perhaps more fundamentally it involves a confusion between a local and a global description of a sphere.]

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How do you get £20 from an Irishman? Ask him to lend you £40. Then say, “Give me £20. Then you’ll owe me £20, I’ll owe you £20 and we’ll be all square.”

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man who got the longer sentence said to his cell mate, “ You take the bed nearer the door, since you ’ll be released first.” [Presumably, if the periods of imprisonment were five minutes and ten minutes, the joke would not be funny. Are there periods of time at which the joke begins to be funny? Is this a joke about differences and orders of magnitude?]

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An Irishman was charged with murder and was sent for trial by jury. To every one’s surprise he pleaded guilty. Nevertheless the jury returned a verdict of “ Not guilty.” “How on earth have you reached a verdict like that?” asked the judge. “The man pleaded guilty.” “ You don ’t know him like we do, your honour , ” said the foreman of the jury. “He ’s the biggest liar in the country and you can’t believe a word out of his mouth.”

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Mathematical and Ethnic Jokes

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An Irishman who kept all his money under the mattress was asked why he didn t ’ keep it in the bank on account of all the interest he would receive. “I’ve worked

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all that out ,” he replied . “I put a little away every week for the interest too.” “ Hello, is that Dublin double two, double two, two?” “ No, you ’ve got the wrong number. This is Dublin two double two double two.” “Sorry for troubling you in the middle of the night.” “That ’s all right, I had to get up anyway because the phone was ringing ” . [The associative law or its absence is responsible for quite a bit of humour and ambiguity, e.g. “help blind children ”,“ black men’s socks” and an item I once saw on a menu , “ half fresh grapefruit.”] “ Hello, is that 999?” “ No, this is 998.” “ Well , would you pop in next door and tell them my house is on fire?” Someone was explaining to an Irishman how nature sometimes compensates for a person ’s deficiencies. “For example,” he told him , “ if a man is deaf , he may have keener sight, and if a man is blind, he may have a very keen sense of smell.” “ I ihink I see what you mean ,” said the Irishman. “I ’ve often noticed that if a man has one short leg, the other leg is always a bit longer.” [ Is this simply a > b b < a?] Have you heard about the new Irish Sweepstake with a first prize of a million pounds? It ’s one pound a year for a million years. An Irish clock tower had four clocks each showing a different time. When a tourist was foolish enough to ask for an explanation he was told quite logically , “ If they all showed the same time, we would need only the one clock.” Irish mathematicians are very good at computations because they work with watches which are an hour fast and ten minutes slow. It used be said of the West Clare railway in Ireland that time tables were provided only to show how late the trains were. [ Axes of co-ordinates?] An Irishman in a slightly inebriated condition got on a bus one evening and said to the conductor, “ Did you see me get on the bus just now ?” “Yes,” said the conductor. “Have you ever seen me in your life before ? ” “ No,” said the conductor. “Then how did you know it was me?” said the Irishman . [This joke probably defies logical analysis]. An Irishman asked his barber for something for his hair and was given a bottle of conditioner. “How much of this should I put on?” he asked. ‘The less you use of it the better,” said the barber. [ I’m convinced this is a joke about open subsets of [0,1]. ] An Irishman went to the doctor and complained he had severe pains in his feet. The doctor took one look at him and said , “I can see what is the matter with

Comic Sections

84

you - you’ve got your shoes on the wrong feet.” “But doctor,” protested the Irishman,“these are the only feet I’ve got.” [What better way to illustrate the difference between permutations and combinations!] An Irishman went to London for a week ’s holiday. He wrote home on a postcard as he left: “Having a wonderful time here it only rained twice - the first time for three days and the second time for four days." [Even 3 + 4 = 7 can be funny if placed in the right context .]

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What do you find on an Irish Klein bottle? “Open other end ”. One Irishman asked another why the days in summer were longer than the days in winter “Heat expands, while cold contracts," explained the other. [This is a good example of a situation where the statements are all correct but the conclusion is not a logical consequence of the hypothesis causality and an argument with truth value one are not equivalent? Ian MacDonald in his excellent Theory of Groups asks if the following is true or false: All groups of order 13 are abelian and all groups of order 17 are abelian . But 15 is the mean of 13 and 17 so all groups of order 15 are abelian.]

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One Irishman was complaining to another that he had gone on an expensive fishing trip and caught only one fish. “That fish,” he told him, “cost me £500.” “Weren’t you lucky,” said the friend, “that you didn’t catch two.” An Irishman wanted to become a great actor so pestered producers for parts despite his lack of acting ability. Finally his persistence was rewarded, and he was given a very small part in which he had to walk onto the stage and in answer to the question “Is it raining?” reply “It is,” and then leave the scene. Our actor practised the line for months with every possible intonation and variation and drove all his friends around the bend saying “it is”, “it is”, “it is” at every available opportunity. When the big night came, he walked onto the stage, full of confidence, and said “Is it?” [For those of you reading this book from front to back this will be the second joke you will have read about permutations on two objects. For those of you reading from back to front, and there are many, there is a treat in store if your feet are sore.] A very old Irishman was heard to remark that if only he could get through the month of March, he usually found that he lived until the end of the year. An Irishman once captured a leprechaun and demanded three wishes in return for letting him go. “I’ll give you two wishes,” said the leprechaun, “because agreeing to grant you three wishes was your first wish.” (The leprechaun’s first name was Russell.) “Okay,” said the Irishman, “I wish for a purse full of gold, so that no matter how many golden coins I remove from it , there will always be some

Mathematical and Ethnic Jokes

85

left.” “ Done,” said the leprechaun, and produced the purse. The Irishman tried it out and was delighted [how did he know it wasn’t a large but finite number of coins?]. “What about your final wish?” said the leprechaun. “That’s a wonderful purse entirely,” said the Irishman. “I wish I had another just like it.” [This joke shows a very clear understanding of the fact that the cardinality of the union of two countable sets is the same as the cardinality of a countable set. I wonder if perhaps Cantor met this leprechaun? Personally I would have wished for as many wishes as I liked as my second wish, or is this just wishful thinking?] Having ordered a pizza, an Irishman was asked whether he would like it cut into four or eight pieces. “Better make it four,” he replied. “I don ’t think I would ever manage to eat eight pieces.” An Irish farmer was asked what sort of a year he had on his farm. “ Average," he replied. “Worse than last year but better than next year is going to be.” An Englishman, a Scotsman and an Irishman each put an odd number of spoons of sugar in his tea. Together they put 16 spoons of sugar in their tea. How

come? The Englishman put in one spoon, the Scotsman put in one spoon and the Irishman put in 14 spoons, which is a very odd number of spoons in a cup of

tea. .

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An Irishman once visited a physics laboratory and saw an inverted image of himself in a large concave mirror. His explanation was that the mirror had been hung upside down.

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A SHORT DICTIONARY OF

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9. A straightforward calculation g i v e s. me the best part of a week , gives. . .

A very difficult calculation, which took

10. Without loss of generality: I can’t handle the general case at all. 11. Which completes the proof : Which completes the proof , hope I 12. Evidently , clearly , obviously : Maybe. 13. Using a deep result of Vernon: Vernon ’s work is completely beyond me, but I know a useful theorem when I see one. 14. An interesting comparison might be made between the present results and those of Barry: There is no connection at all, but his name looks great in my refer-

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As a young graduate student I was frequently peiplexed by certain words and phrases which cropped up again and again in the research papers which I attempted to read. Conventional mathematical dictionaries gave me no help whatsoever, but experience has since taught me the true meaning of many of these expressions. For the benefit of those who find themselves in the same position, I offer a selection , in the hope that it will stimulate others to contribute to this sadly neglected area of mathematical education. 1. The proof is left as an exercise : I’ve lost the envelope on which I jotted this down , but it seemed reasonable at the time.

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2. While the results of Holland are relatively deep: Holland once mentioned a paper of mine in his references.

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5. I wish to thank the referee for a number of useful suggestions: The old meanie cut me down from twenty pages to a miserable four. 6. While only partial results have been obtained: I’ve made no progress at all with this problem but I figured I could get at least one publication from it. 7. It is well known that : I’m not quite sure how to prove this but I’m hanged if I’m going to go to the trouble of finding out who first discovered it.

8. Harte ( oral communication ) has shown that : I cornered him and bored him to tears during a coffee break at a recent conference. i

15. On pseudo compact semiheaps with involution I : I hope to get at least four papers out of this useless and obscure topic. 16. This problem is of great theoretical significance ’ : I m the only one who is interested in it or knows anything about it.

17. I wish to thank Miss O’ Keeffe for her patience and excellent typing : She has threatened never to type another word unless I put this in.

18. I wish to thank Dr . Seda for some valuable suggestions: All the ideas and work are due to Dr. Seda, my supervisor

.

19. It is natural to ask the question: One of my research students just has. 20. We are sorry that due to lack of space we cannot publish your article : Some people have a neck sending such rubbish to a distinguished journal like ours. 21. Some of his results are in conflict with ours: The guy’s crazy: SL(2,p) is an obvious counterexample. 22. Dr . Fitzpatrick has kindly pointed out an error in Lemma 3: Why doesn ’t that ****** mind his own business? 23. I wish to thank my supervisor for his valuable assistance in the preparation of this paper: I saw him once in the distance at a conference .

APPENDIX: QUESTIONER ’S HANDBOOK Recently I attended a mathematical lecture given by a guest speaker where absolutely nobody, except possibly the speaker, had the remotest idea what was going on. Normally one can absorb at least some of the preliminary definitions and follow (say) the first blackboard -full of development of the theory, but on this occasion everyone was completely lost after the first definition . After the speaker

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had finished over an hour later to an enthusiastic round of applause, the chairman asked for questions, and of course there was a deathly and highly embarrassing silence. There and then I resolved to put together a collection of universal questions for use in such situations. Such questions must sound sensible, but they are designed to cover up the total ignorance of the questioner rather than to elicit information from the speaker. The following is the list I came up with:

1. Can you produce a series of counterexamples to show that if any of the conditions of the main theorem are dropped or weakened then the theorem no longer holds? The speaker can almost always do so if not you may have presented him with a stronger theorem.

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2. What inadequacies of the classical treatment of this subject are now becoming obvious?

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3. Can your results be unified and generalised by expressing them in the language of category theory? The answer to this question is always NO. 4. Isn' t there a suggestion of Theorem 3 in an early paper of Gauss? The answer to this question is almost always YES.

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5. Isn’ t the constant 4.15 in Theorem 2 suspiciously close to 4 JT / 3? This question can clearly be generalised for any constant k “Isn’t k suspiciously close to ( p/ q )ir (for suitable integers p and q )T’

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7. Are you familiar with a joint paper of Besovik and Bombialdi which might explain why the converse of Theorem 5 is false without further assumptions? This is a tricky question to ask unless you like living dangerously The answer is always “NO” unless the speaker is playing the same game as you are, because Besovik and Bombialdi do not exist, and even if by some unfortunate chance they do exist, it is very unlikely that they have written a joint paper. If the speaker calls your bluff and asks for details and a reference, tell him the paper is available only in Albanian with Portuguese summaries. Promise to mail him a copy but forget to do. Notice, however, that even this precaution may not get you out of all trouble. The material in this appendix was originally published in article form in the American Mathematical Monthly. Shortly afterwards the editor, Paul Halmos, sent me a copy of the following letter received from two outraged correspondents: Groundhog Day 1983 Dear Dr. Halmos, We have never been so insulted as when we read on pp. 42-44 of volume 90 of the Monthly that we do not exist. We are shocked that a reputable journal regards this and the

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subsequent comments as an acceptable

way of criticising MacHale regards our existence as an “unfortunate chance ”our joint papers. If Desmond , that is his loss, not ours .

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Igor N. Besovik Professor of Proletarian Mathematics Gulag University Chuguyevka, Siberia CCCP

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Antonioniottfd Bombialdi Archimedean University of Sicily “Give Us a Fulcrum and We Will Move the Red Brigade” Syracuse

8. Why not get a graduate student to perform the horrendous calculations mentioned in Theorem 1 in the case n 4? The answer is always, “I’ve a = student doing just that at the moment ”

.

9. Could you draw a simple diagram to show what the situation looks like for n = 2? Be careful that he hasn’t already done

so.

10. What textbook would you recommend for someone who wishes to get students interested in this area? The speaker has almost invaria bly written such a textbook himself and will be delighted you asked this questio n. If he hasn’t then you can ask the next question.

11. When can we expect your definitive textboo k on this subject? 12. Why do you think there was such a flurry of activity in this area around the turn of the century and then nothing until your paper of 1979? There is no danger of your getting the true answer here, that people in the period in between had more sense.

In general, a good ploy is to stop halfway through a meaningless question you are asking and pretend you have suddenly seen the answer yourself. However, never, never ask 13. What are the applications of these results ? The speaker is probably embarrassed enough already.

Mathematical Graphiti

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Sign near a level crossing: The average time it takes a train to pass this crossing is seven seconds - whether your car is on it or not.

I’m not worried about losing my job because of computerisation. They still haven ’t invented a machine that does what I do - absolutely nothing.

MATHEMATICAL GRAPHITI

I ’m not quite sure what the uncertainty principle is. The theoiy of relativity in a nutshell - how long a minute is depends on which side of the bathroom door you are. t

Rubik ’s cube cures sanity.

Statistics means never having to say you’re certain.

To every complex mathematical problem there is a simple solution wrong.

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Every Ph.D. thesis just transfers the bones from one graveyard to another. Smoking is the principal cause of statistics. Computers are fantastic - in a few seconds they can make a mistake that would take a thousand people working for millions of years to equal.

Complete set of Bourbaki for sale. Never used. My research students know everything.

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A pure mathematician is someone who has found something more interesting then sex.

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To err is human but to really foul things up you need a computer.

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Mathematics departments are storehouses of knowledge. All the students bring a little in, none of the graduates takes any away, and the stuff naturally accumulates.

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Don’t let pessimistic statistics about the future worry you. Remember in 1850 it was predicted that if the traffic kept increasing at the same rate the entire surface of the earth would be covered in six feet of horse manure by 1970. A pure mathematician is like a lighthouse in the middle of a bog - brilliant but

useless.

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What mathematics needs is fewer people telling us what mathematics needs. Heisenberg may have been here.

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If God wanted us to decimalise why were there twelve apostles?

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Absolute zero rules 0°K. Time flies like an arrow - fruit flies like an apple. The latest thing in computers is almost human blames another computer.

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My girlfriend ’s vital statistics are 42-24-38, but unfortunately not in that order. Please do not cross the railway lines as it takes ages to untangle them again. The laws of the Catholic church on contraception forbid the use of physics and chemistry but allow and encourage the use of mathematics. [ Does the logarhythm method lead to exponential population growth?]

Topologists - where do all the holes go when Swiss cheese is eaten?

To say something clever - think of something stupid and say the opposite. Constipated mathematicians work it out with logs.

To every mathematical problem there is a solution . The difficulty lies in finding it. Rubik started the age of cuberty.

Never attribute to deliberate malice that which can be adequately explained by straightforward stupidity. The foundations of mathematics are illogical - and I can prove it.

I ’ve told you a million times not to exaggerate.

- we are at

Slide rules O.K.

Those who can, do. Those who can ’t, teach. Those who can ’t teach, teach the history of mathematics.

On a mathematical plagiarist: many of you will have read his papers; some of you will have written them .

The new mathematics is so simple only a child could understand it. The law of the excluded middle either rules or does not rule O. K . i

A topologist is someone who doesn’t know the difference between a doughn ut and a cup of coffee. An applied mathematician is someone who is prepared to use mathem atics with out any foreplay.

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Research in mathematics is like making love to an elephant - there’s no pleasure in it, you run the risk of injuring yourself, and it’s years before anyone can see the results.

Many a conjecture is just a theorem in waiting for the right proof to come along. Problems in mathematics increase in geometric progression but solutions to them increase in arithmetic progression. Any theorem that takes over fifty words to state contains at least one flaw. Any proof longer than four pages contains at least one mistak . e

Is a lemma that proves two results a dilemma? Infinity is far out.

If a problem in mathematics looks easy it ’s probably hard; if it looks hard it’s probably damn near impossible.

Save energy - ride a Turing machine.

The law of conservation of difficulties: if a result is deep there is no easy way of proving it.

A mathematics lecture is a process whereby the notes of the lecturer become the notes of the student without passing through the minds of either. Any graffito with more than eight words is false. Have you heard about the extinct tribe of Red Indians who mastered vector resolution?

Anything you prove will have been proved by some other son of a bitch last month despite the fact that nobody else has looked at the problem for fifty years.

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Peano counts on his fingers. Church ’s thesis was referred.

[Life is an integral function of happiness over the time between birth and death.] The most productive time time to proof read a paper is just

a.

Bertrand Russell didn ’t know what he was talking about. Weierstrass failed his M-test.

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Don ’t expect to make any serious mathematical discoveries until you are over fifty. It takes that long to get over the distractions of sex.

Some series are fourier than others.

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The great tragedy of mathematics is a beautiful conjecture ruined by an ugly fact.

The probability of a piece of toast covered in marmalade falling face downwards on a carpet is directly proportional to the cost of the carpet. [Exercise: show that this statement covers a flaw ] .

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Down with non-standard Lebesgue measure theory. Descartes thought he was here.

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Mathematics without the axiom of choice is like a fish without a bicycle . 2 2 » = j = k? = ijk = 1

Enjoy the rapture of the first few moments after you’ve proved a mathematical theorem - before you find the mistake.

Never trust any mathematical result you prove after 11 p.m. The set of all sets which do not contain themselves as members is a communist plot.

The principle of Occam’s razor shave himself.

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If it wasn’t for Newton we would all be eating bruised apples. Godel’s theorem cannot contain a proof of its own consistency.

- Bertrand Russell shaves everyone who doesn’t

The probability of cutting yourself while shaving is directly proportional to the importance of the event for which you are shaving.

Mathematical research means reading three papers that have never been read before in order to write a paper that will never be read again .

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Mathematical problems are either easy or impossible. I wish that mathematical students who have difficulty in commu nicating would just shut up about it.

Definition of a mean mathematical examiner - he deducts a mark you if don’t add an arbitrary constant when integrating but gives no mark for verifyin g that the theorem is true for n = 1 in an inductive proof.

Service courses on the application of mathematics should be banned. After all, you don ’t see courses for people who want to swim with waders on.

Statistics are like a bikini. What they reveal is suggestive but what they conceal is vital.

I don’t have any solution but I certainly admire the problem. If God is perfect why did he create discontinuous functions? Written underneath He didn ’t, Man did.

A right derivative is neither partial or impartial. We use statistics as a drunken man uses a lamp-post illumination.

A page from Einstein’s notebook : .

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Statistics can be used to support anything - especially statisticians. Is this a question ?

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Any joke sounds funny the way the head of the department tells it. Just because I have a Ph.D. in mathematics doesn’t mean I’m stupid

Statisticians have all the standard deviations. NOTICE: We shoot every tenth trespasser - the ninth one has just left. If you think you ’ve got the solution to a difficult mathematical problem, the question was poorly phrased .

A statistician is someone who can have his head in the fridge and his feet in the fire and claim that on average he’s feeling fine.

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Inside every big computer program there’s a small computer program trying to get out.

E = jrjfi ] , E = frfip ] , E = me2 Mathemetics make me sick. Trigonometry is when a man marries three wives at the same time . Applied mathematicians have moments of inertia. A number is just a castrated statistic. The probability of life on earth having started by accident is about the same as that of the Oxford English Dictionary resulting from an explosio n in a printing shop. Kirkman had a problem with schoolgirls. Mathematical analysis is a subject where you spend half of the time proving x > y and the other half proving y > x , when all along you know x = y. There is no such thing as a free group.

When drawing a graph the thickness of the curve is inversely proportional to the reliability of the data.

- more for support than for

Never say never.

I’ve just built the world ’s largest microchip. The number of first class honours awarded in mathematics is proporti onal to the number of glasses of wine the external examiner has for lunch. The ratio of horses’ extremities to horses is always greater than one. Almost everybody has more than the average number of arms. I’m not interested in the largest known prime I want to find the largest known integer.

-

Why does nobody study sets which are finite almost everywhere ? Mathematicians who lack professorial timbre are not chair material. The only sure thing about a safe bet is the uncertainty. Have you heard about the binary millionaire? He owned £64.

Conic sections have their eccentricities.

How come abstract algebra is harder than concrete algebra? How about this for a contradiction ? Quarter of a circular pie is in fact pi over

two.

The main difference between me and a millionaire is that he’ s working on his second million while I’m working on my first.

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The relationship between pure and applied mathematics is based on trust and understanding: the pure mathematicians don’t trust the applied mathematicians, and the applied mathematicians don’t understand the pure mathematicians.

2 + 2 = 5 for large values of 2.

Mathematicians spend 90% of their time thinking of mathematics, sex and food. If mathematics lets them down they turn to sex and food in turn. That is why there are so many fat mathematicians.

A theorem is the maximum weight the hypothesis will bear. ( TT 4 + 7r6 ) e = 2.718281809... you can have your pi and e to it too. I stink therefore I am.

plated.

I hear and I forget , I see and I remember, I do and I understand. A category theorist is someone who will solve a problem you didn’t realise you had in language you can ’t understand . Some students in mathematics are now so tense they can hardly go to sleep during lectures.

Will humans ever replace computers? Nobody knows what he cannot do until he tries.

!

In mathematics one should always work from the known to the unknown or vice

versa.

Logicians may not always be right , but they are never wrong. How dare you presume I ’m an analyst.

Deduce is what you get when you squeeze de orange. Mathematics books are full of odd -looking apparently meaningless characters like any college mathematics department. Motto of the computer: garbage in , garbage out.

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The difference between genius and stupidity is that genius has its limits. Motto of the applied mathematician: it ’s all very well in practice, but how does it work in theory?

Never look for the fifth root of a cat. [Shurely some mistake here? ]

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Every degree in mathematics should include a course in category theory just as every degree in medicine should include a course on venereal diseases.

The dilemma of the mathematics teacher: I know you believe you understand what you think I said, but I am not sure you realise that what you think you heard is not exactly what I meant.

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A mathematical tutor is someone who spends his time converting genuine thirds into doubtful seconds.

Pelvic geometry is the only sound basis on which a marriage should be contem

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8 SOME REVIEWS FROM THE JOURNALS

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standard of refereeing of papers. I began to launch forth on the unpredictibility of referees, observing that a recent paper of mine had been published though it was quite unworthy of publication and that I had since discovered that the main result was known already. “ Before you proceed any further,” said my listener,“I think you should know that I refereed that paper.” A Japanese author is reputed to have written the following in a letter to the editor of a journal: Please thank the referees for their many suggestions: I would like to execute them .

Not alone is the principal (and only) theorem of this paper absolute ly false, but the example quoted to illustrate it is in fact a counterexample

.

This paper purports to prove a number of results many of which are well -known and most of which are false. This paper contains many new and interesting results. Unfortunately, the results that are interesting are not new and the results that are new are not interesting. In joint papers , authors are often reluctant to disclose which author has proved what. This praiseworthy custom was taken to the limit in the case of a joint paper on mathematics which carried the following footnote: Since writing this paper, one of us has died. This book creates a much-needed gap in mathematics. Most papers are rejected by journals not because the referee can ’t understand them but because he can. The ones the referee can ’t understa nd are usually

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A pretentious young research student once submitted to a Chinese journal a paper entitled “On the Cohomology of Regular Compact Semihea ps with Bounded Linear Pseudo Involution I ." He received the following reply from the editor : We have read your manuscript with boundless delight. If we were to publish your paper it would be impossible for us to publish in the future any work of a lower standard. As it is inconceivable that we shall ever again see its equal, we are, with great regret, compelled to return your exquisite paper and we beg you to overlook our shortsighted timidity. The surprising thing about this paper is that a person who could write it, would. I once had a very embarrassing experience at a large mathem atical conference. At breakfast I got into conversation with a veiy pleasant gentleman sitting beside me and we introduced ourselves. The conversation turned to journals and the

-

There is less in this paper than meets the eye.

From a review of a book called “Ignorance in Mathematics”: The author has contributed greatly to his subject. This lemma, which is trivial , is due to Hardy and Littlewood.

The following extract from a published paper is absolutely genuine and l can vouch for it. Only the names have been omitted to protect the guilty: Professor X has kindly drawn my attention to the fact that some of the main results of [1] appear in his paper [2], Even worse, I discovered when I checked the review of this paper that I myself had written the review for Mathematical Reviews about five years ago. Worse still , it is apparent from X’s paper that my theorem 2 is incorrect and renders the analysis in the second half of my paper incomplete. However, the theorem can be corrected, and it seems worthwhile doing this since it leads to. . .

This is a seventeen-page proof of Fermat’s Last Theorem which the author claims requires powerful mental resonance in order to understand. Clearly, the author’s mind is in a state of resonance which the reviewer was not able to attain. The author discusses valueless measures in pointless spaces. This paper gives wrong solutions to trivial problems. The basic error, however, is not new. The main theorem proved in this paper is correct. However, only the identity transformation acting on the trivial space satisfies the hypothesis.

A reviewer was asked by an aspiring young mathematician if he thought the young man’s most recent paper was good. “My dear fellow,” he replied, “good isn ’t the word .” Everybody knows him as the author of the most successful book on differential equations that has ever appeared in any language; although it was first published as long ago as 1885, it is still being reprinted. I would venture the opinion that

his work has done more than anything else to retard the true development of the subject; for over two generations it has continued to put wrong ideas into people ’s heads concerning the nature and scope of the theory, and thanks to the author’s forceful and authoritative style, in this it has been overwhelmingly successful. Y ur sins mil find you out. The following is an algebraised version of a comment which appeared in a mathematical magazine : X notes that the proposer stated the same problem in Y. The proposer also submitted part of the problem to Z and it appeared there as problem W in T. The conclusion of an actual review, scathing in the extreme, of a mathematical textbook: The printing and binding of this book are excellent.

9 A FEW RIDDLES

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What is a kittegory? A small category.

What is yellow and equivalent to the axiom of choice? Zorn’s Lemon. What is purple and commutative? An Abelian grape. What is the difference between a part-time professor of mathematics and a fulltime professor of mathematics? A part -time professor of mathematics is away part of the time while a full-time professor of mathematics is away all of the time.

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When is 7r rational? [Sadly, an authentic extract from a school textbook ] When n = S'1,

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“ the law of the excluded widdle.” I think this is perhaps an improvement on the original.]

[ When I put this riddle to Phil Rippon he replied

When two cats are sitting on an inclined plane, which one will fall off first? The one with the smaller /z.

Have you heard about the alcoholic mathematician? He was awarded the W.C. Fields medal for drinking out of a Klein bottle.

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What did they say to Archimedes before he had a bath? You reeka.

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How does a statistician count sheep? He counts their legs and divides by four.

Q. Is Volkswagen pronounced with an initial V? Q. What is the shortest distance between two points?

How do you know if an algebraist ’s bicycle (two-cycle) is defective? If it satisfies the descending chain condition but not the ascending chain condition.

What is the difference between an ordinary actuary and a Sicilian actuary? An ordinary actuary can tell you very accurately how many of a given population will die in the coming year, while a Sicilian actuary can tell you their names and addresses.

What is the normal mode of a vibration? Over and back.

Have you heard about the fellow who had a joint degree in mathematics and art? He got a job painting computers.

What goes oink A pork pie.

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Have you heard about the girl who failed her examination on co-ordinate geometry because she spent all her time at the races? She put de horse before Descartes.

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What is a net? It’s a load of holes tied together with string.

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Why is it called mathematical physics? It ’s called mathematical physics, because it has nothing to do with either mathematics or physics.

-

What do you give a lady topologist for her birthday? A compact set.

What is the difference between an applied mathematician and a pure mathemati cian ? An applied mathematician has a solution for every problem while a pure mathematician has a problem for every solution.

What had Archimedes discovered when he shouted “Eureka” in the bath ? The soap.

Which is the oddest prime number? Two, because it is the only even prime number.

Q. Two statisticians, one veiy fat and the other very thin, fell off the top of a high building. Which one hit the ground first? A. Who cares, so long as they both are statisticians.

What sort of computer did Isaac Newton use? An Apple.

What is the definition of a careful author of a mathematics textbook ? Someone who apologises for numbering the sections before he has defined the natural numbers.

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Question: If one Mona Lisa is worth £50 million, how much are two Monae Lisae worth?

What is a number? Number is the answer to the question “How many?” [Taken from a quaint old textbook.]

Is research in mathematics difficult? No, it’s either easy or impossible. A number of years ago there was a craze for inverted riddles where you were given the answer and you had to guess the question. Here are a couple of examples. A. 9W A. Mickey Rooney. The corresponding questions were

How many great mathematicians are there in the world today? One less than you think. What does a retired mathematician call his house? Aftermath. What is the value of the product

(i - a ) ( x - 6) ( x - c) • •• (x - z )? Answer 0.

What did the kernel say when it grew up? Geometry.

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It is true that a mathematician who is not also something of a poet will never be a perfect mathematician. - Karl Weierstrass

MATHEMATICAL WIT AND WISDOM

The mathematician’s patterns, like those of the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. There is no permanent place in the world for ugly mathematics. - G .H . Hardy

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it gives the impression of also being beautiful. - George Boole One of the aims of this book is to humanise mathematics, and one of the best ways of achieving this is to listen to what has been said about mathematics both by those in the subject and those outside it. Not all of the quotations are profound (note that a down-market way of saying this is ’’all of the quotations are not profound”), in fact some are just the result of people trying to be clever or impressive, but most of them are worth thinking about. If you regard yourself as a serious or profound commentator on mathematics and you are not quoted in this chapter, then you are clearly mistaken and nobody has been listening to what you have been saying all these years. But do keep trying; after all , this book could run into further editions. On the other hand , you may find yourself quoted in this chapter and have no recollection whatsoever of having uttered the gem attributed to you. If that is the case, be assured that I was there and I heard you.

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There are two classes of people in the world , those who divide people into classes and those who don ’t. - Robert Benchley

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- Isaac Newton

How happy the lot of a mathematician. He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. - W .H . Auden

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The Analytical Engine weaves algebraic patterns, just as the Jacquard loom weaves flowers and leaves. - Ada Lovelace The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them. - George Polya

Mathematics possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture. - Bertrand Russell

-

Why should I refuse a good dinner simply because I don ’t understand the di gestive processes involved? - Oliver Heaviside [ when criticised for his daring formal use of operators before they could be justified logically ]

Mathematicians are like lovers. Grant a mathematician the least concession , and he will draw from it a consequence which you must also grant him , and from this consequence another. - Fontenelle

I drink therefore I am. - W .C . Fields

Let no one ignorant of mathematics enter here.

- Plato

When you have eliminated the impossible, whatever remains, no matter how improbable, must be the truth. - Sherlock Holmes

Mathematical creativity is terrific, but plagiarism is much faster.

- Anon

This series is divergent, therefore we may be able to do something with it. Oliver Heaviside

God over geometries. - Plato Each of the positive integers is one of my personal friends.

Can someone please tell me what is a Hilbert space?

- J .E . Littlewood

- David Hilbert

Cowboys have a way of trussing up a steer which fixes the brute in such a way that it can neither move nor think. This is the hog tie, and it is what Euclid did to geometry. - E .T . Bell

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Mathematics is the only science where one never knows what one is talking about nor whether what one says is true. - Bertrand Russell

Those who work in number theory are like lotus-eaters: having once tasted this food they can never give it up. - Leopold Kronecker I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh. - Thomas Mann The first test of potential in mathematics is whether you can get anything out of geometry. - J .E . Littlewood God is subtle but he is not malicious. - Albert Einstein If our brains were any simpler, we would be too damn stupid to understand them. - J .G . Thompson Algebra is generous: she often gives more than is asked for. - J . D' Alembert

And what are these fluxions? The velocities of evanescent increments? They are neither finite quantities, nor quantities infinitely small , nor yet nothing. May we not call them the ghosts of departed quantities? - Bishop Berkeley

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Mathematics is the door and the key to the sciences.

Detest it as lewd intercourse; it can deprive you of all your leisure, your health , your rest, and the whole happiness of your life. - Wolfgang Bolyai [in a letter to his son, advising him to give up his attempts to prove the Euclidean parallel

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God made the integers, all else is the work of man.

- Leopold Kronecker

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and it immediately becomes something different. Goethe

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To the student there is no part of elementary mathematics so repulsive as is spherical geometry. - P .G . Tail The essence of mathematics lies in its freedom. - G . Cantor

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What is algebra exactly? Is it those three cornered things? - J .M . Barrie

Mathematics is facts; just as houses are made of stones, so is mathematics made of facts; but a pile of stones is not a house and a collection of facts is not necessarily mathematics. - Henri Poincari. Mathematics is the music of reason.

- J .J . Sylvester

I spent all day working on a new treatment of Euclidean geometry. In the morning I took an axiom out , and in the afternoon I put it back again. - Oscar Tame A proof tells us where to concentrate our doubts. Mathematics is a collection of useful recipes.

- W .H . Auden

- Paul Valery

- Albert Einstein I can speak mathematics like a native. - Spike Milligan I don’t believe in mathematics.

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In ancient times they had no statistics so they had to fall back on lies. Leacock

When a philosopher says something which is true then it is trivial. When he says something that is not trivial then it is false. - C .F . Gauss

Applied mathematics will always need pure mathematics just as anteaters will always need ants. - Paul Halmos

The mathematician has reached the highest rung on the ladder of human thought. - Havelock Ellis

Oftimes Archimedes’ servants got him against his will to the baths , to wash and anoint him; and yet being there , he would ever be drawing out of the geometrical figures , even in the very embers of the chimney. And while they were anointing of him with oils and sweet savours , with his fingers he drew lines upon his naked body ; so far was he taken from himself , and brought into ecstasy or trance , with the delight he had in the study of geometry. - Plutarch

Population, when unchecked, increases in a geometric ratio. Food to support that population only increases in an arithmetic ratio. - Thomas Malthus A good mathematical joke is better, and better mathematics, than a dozen mediocre papers. - J .E . Littlewood Geometry is the art of correct reasoning on incorrect figures.

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- George Polya

Aristotle could have avoided the mistake of thinking that women have fewer teeth than men by the simple device of asking Mrs. Aristotle to open her mouth. Bertrand Russell

If I have been able to see further than others, it was only because I stood on the shoulders of giants. - Isaac Newton

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Everything is vague to a degree you do not realise till you have tried to make it precise, and everything precise is so remote from everything that we normally think, that you cannot for a moment suppose that it is what you really mean when we say what we think. - Bertrand Russell Probability is the only branch of mathematics in which good mathematicians frequently get results which are entirely wrong. - C .S . Pierce

There is no royal road to geometry. - Euclid

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Mathematics is the science which uses easy words for hard ideas. - Kasner and Newman The moving power of mathematical invention is not reasoning but imagination.

- Augustus de Morgan

Again, a touch of humour (strange as the contention may seem) in mathematical works is not only possible with perfect propriety, but very helpful. - G .H . Minchin

Science is an exercise of the human brain to grasp the principles by which the universe works and, to write them down , if possible, in crisp, precise, mathematical terms. - Magnus Pyke I could prove that God exists statistically.

There are mathematicians of the first order who cannot count. - Novalis

- George Gallup There are three kinds of lies: lies, damned lies and statistics. - Benjamin Disraeli A little inaccuracy sometimes saves tons of explanation. - H .H . Munro ( Saki ) [

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This principle is indispensable in the teaching of mathematics.] “Can you do addition?” asked the White Queen. “What is one and one and one and one and one and one and one and one and one and one?” “I don’t know,” said Alice, “I lost count.” Lewis Carroll The Council of the Royal Society is a collection of men who elect each other to office and then dine together at the expense of this society to praise each other over wine and give each other medals. - Charles Babbage There was far more imagination in the head of Archimedes than in the head of Homer. - Voltaire

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I have never done anything “useful.” Judged by all practical standards, the value of my mathematical life is nil. - G .H . Hardy The mathematician may be compared to a designer of garments who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity of clothing such creatures, but this was long ago. To this day a shape will occasionally appear which will fit into a garment as if the garment had been made for it. Then there is no end of surprise and delight. Dantzig

Don ’t talk to me about mathematics; I ’ve come to the conclusion that I can live without it. - Prince Philip , Duke of Edinburgh

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Very little of mathematics is useful practically, and that little is comparatively dull. - G .H . Hardy [To which Frederick Soddy replied: From such cloistral clowning the world sickens.]

There is a special room in hell for statisticians. It is filled with typewriters and monkeys. Every time a monkey walks on one of the typewriters, it types by chance one of Shakespeare’s sonnets. - Bertrand Russell If a man ’s wit be wandering, let him study mathematics. - Francis Bacon As a net is made up of a series of ties, so everything in this world is connected by a series of ties. If anyone thinks that the mesh of a net is an independent isolated thing, he is mistaken. Each mesh has its place and responsibilities in relation to other meshes. - Buddha A single room is that which has no parts and no magnitude. - Stephen Leacock Algebra goes to the heart of the matter and it ignores the casual nature of par ticular cases. - E C . Titchmarsh

Mathematicians who are only mathematicians have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise they are inaccurate and insufferable, for they are only right when the principles are quite clear. - Pascal There is no national science, just as there is no national multiplication table Chekhov

Any two meals at a boarding house are together less than two square meals. Stephen Leacock

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A man has one hundred dollars and you leave him two dollars: that’s subtraction. - Mae West Although this may seem a paradox , all exact science is dominated by the idea of approximation. - Bertrand Russell

The study of exact sciences is not so good a discipline as is commonly supposed for preparing the mind against inaccuracies of thought and expression in matters full of darkness and pitfalls. I have seen many illustrations of this in the arguments of mathematicians when out of their element. - Chauncey Wright A serious and good philosophical work could be written that would consist entirely of jokes. - Ludwig Wittgenstein I would rather discover one scientific fact than become King of Persia.

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Point set topology is a disease from which the human race will soon recover. Henri Poincari

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I could be bounded in a nutshell and count myself a king of infinite space. Hamlet [Perhaps he had been reading Measure for Measure .]

What use are socks? They only produce holes.

- Albert Einstein

“For example” is not proof. - Yiddish proverb

Computers are important, but not to mathematics.

A Unitarian is someone who believes in at most one God. - A.N . Whitehead [Perhaps Unitarians are correct. Here is a proof that there exists at most one God . If there is no God, we are done, in every sense of the word . Assume therefore that there is a God . Now God is everywhere, so there is no place for another

A problem worthy of attack Proves its worth by hitting back.

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as required.] The function of the lecturer is to expound a few classic documents and to hand down as large and pleasant a store as possible of academic habits, maxims, and anecdotes. - Santayana

Reductio ad absurdum is one of the mathematician ’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the whole game. - G .H .

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The trouble with random floods is that they are so unpredictable. - Spokesman for Dublin Corporation after the disastrous floods of August 1986

All life is six to five against.

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- Nietzsche

I went to Winchester in 1840 at twelve years old , able to solve a quadratic equation, and left it at eighteen, competent to perform the same task badly. - W . Tuckwell

God. Next we show that the existence of God is equivalent to the Axiom of Choice, which is of course equivalent to Zorn ’s Lemma. Consider a partial order of the set of all properties of objects by inclusion . By Zorn’s Lemma, this set has maximal elements. Let God be one of these maximal elements. Now God C God U {existence} so God = God U {existence} and so God exists. Moreover, God is unique. Let God and God ' be Gods. Then God U God' C God , so God D God'; in like manner God' D God , so God = God'. Conversely, given a set {Ai }{i6 4 } of sets, let the unique God pick { ,} e { A,}, * for each i A (God can do anything). Then

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Great men’s errors are to be venerated as more fruitful than little men’s truths.

Moreover, the satellites of Jupiter are invisible to the naked eye and therefore can have no influence on the earth and therefore would be useless and therefore do not exist. - Francesco Sizzi In mathematics, seek simplicity but distrust it. - A .N . Whitehead

A mathematics teacher is a midwife in the birth of ideas.

- George Polya

- Damon Runyon - Paul Halmos

- Piet Hein Pure mathematics consists entirely of assertions to the effect that if such and such a proposition is true of anything , then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it supposed to be true. - Bertrand Russell

On the whole, divergent series are a devilry and it is a shame to base any demonstration on them. - Abel He had heard that one is allowed a certain latitude with widows, so he took the whole 180 degrees. - George Ade

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The sphere of mathematics seems to some the most inhuman of all human activ ities and the most remote from poetry. Yet it is here that the artist has the fullest scope of his imagination. - Havelock Ellis

I am too good for philosophy and not good enough for physics. Mathematics is in between . - George Polya I believe that proving is not a natural activity for mathematicians.

- Reni Thom

Factoring a quadratic becomes confused with genuine mathematical talent.

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They must be true , because if they were not true, no one would have had the imagination to invent them. - G .H . Hardy [Referring to some extraordinary results he received in a letter from the unknown Ramanujan.]

The most boring mathematical concept is tangent bundles.

- George Sanders

Her figure described a set of parabolas thet would have caused cardiac arrest in a yak . - Woody Allen

Mathematics is a dangerous profession; an appreciable proportion of us goes mad. J .E Littlewood

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The book of nature is written in mathematical characters. - Kepler A curved line is the loveliest distance between two points. - Mae West Regard finite group theory as the theory of categories of G-sets for various finite groups G, and then remember that the category of G sets is a Boolean Grothendieck topos which does not satisfy the axiom of choice. - Tomoyuki Yoshida

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