Foolproof: A Sampling of Mathematical Folk Humor Paul Renteln and Alan Dundes

n the discipline known as folkloristics [D1] (the on mathematical culture in general, and even a study of ), a folk is defined as any clue as to the nature of mathematical thinking. group whatsoever that shares at least one As the readership of the Notices is principally common linking factor. The factor could be na- professional mathematicians, we do not feel it is Itionality, ethnicity, religion, or occupation. necessary to explain all the data we present. Most Members of a profession would also qualify as a readers will already be familiar with many if not folk group. Hence, mathematicians constitute a all of the examples given here. That is to be ex- folk. And, like all folk groups, mathematicians have pected. All folklore exists in multiple forms and their own folk speech (slang), , limericks, demonstrates variation. There are often many ver- and , among other forms of folklore. It is pre- sions of a classic or . The same absent- cisely the folklore of a group that defines that minded professor joke, for example, may be at- group. So, mathematicians as a group share a com- tached to various historical individuals, for mon core of mathematical folklore. Some of this example, Norbert Wiener and John von Neumann. folklore tends to be quite esoteric and intelligible So it is likely that each reader will recognize a par- only to members of the group. Outsiders not pos- ticular item even though he or she might have 1 sessing the requisite mathematical vocabulary and heard it (orally) in a slightly different form. knowledge rarely know such esoteric material, and Making Fun of Mathematics even if they did they would probably not under- Many English-language mathematical jokes are stand it. But there is also exoteric mathematical based on word play involving standard mathe- folklore that is known to a limited number of out- matical concepts and terminology. In fact, many of siders, for example, physicists, chemists, and en- the jokes involve food items, which may be a re- gineers. Much of this exoteric folklore consists of flection of the fact that some mathematical con- classic jokes contrasting members of different but cepts are hard to digest, or difficult to swallow: related academic disciplines. We intend to offer a brief sampling of both esoteric and exoteric math- Q: What’s purple and commutes? ematical folklore, concentrating on humorous gen- A: An abelian grape. res such as jokes. We are persuaded that these data not only serve as a basis for identity among 1Many of the following texts come from , mathematicians but also provide a unique window but we have also borrowed freely from various Internet websites, including Paul Renteln is professor and chair of the Department of http://www.xs4all.nl/~jcdverha/scijokes; Physics at California State University, San Bernardino, http://www.math.utah.edu/~cherk/mathjokes.html; and visiting associate in the Department of Mathematics http://www.workjoke.com/projoke22.htm; at the California Institute of Technology. His email address http://users.ox.ac.uk/~invar/links/jokes.html; is [email protected]. and Alan Dundes is professor of anthropology and folklore at the , Berkeley. His email address http://outreach.math.wisc.edu/local/ is [email protected]. miscellany/MathJokes.htm.

24 NOTICES OF THE AMS VOLUME 52, NUMBER 1 There are several other variants of this joke: Q: What is green and homeomorphic to the open unit interval? Q: What’s purple, commutes, and is worshipped by A: The real lime. a limited number of people? A: A finitely-venerated abelian grape. Q: What’s yellow, linear, normed, and complete? A: A Bananach space. Q: What is lavender and commutes? A: An abelian semigrape. Q: What do you call a young eigensheep? A: A lamb, duh! Q: What is purple and all of its offspring have been committed to institutions? Q: What’s the value of a contour integral around A: A simple grape: it has no normal subgrapes. Western Europe? A: Zero, because all the Poles are in Eastern Europe. Q: What’s purple, round, and doesn’t get much for Christmas? Addendum: Actually, there ARE some Poles in West- 2 A: A finitely presented grape. ern Europe, but they are removable!

Q: What’s an abelian group under addition, closed, Q: Why did the mathematician name his dog associative, distributive, and bears a curse? “Cauchy”? A: The Ring of the Nibelung. A: Because he left a residue at every pole.

Q: What’s nutritious and commutes? Q: What is a topologist? A: An abelian soup. A: Someone who cannot distinguish between a doughnut and a coffee cup. Q: What’s hot, chunky, and acts on a polygon? A: Dihedral soup. Q: Why didn’t Newton discover group theory? A: Because he wasn’t Abel. Q: What’s sour, yellow, and equivalent to the Axiom of Choice? Q: What do you get if you cross an elephant and a A: Zorn’s Lemon. banana? A: |elephant|*|banana|*sin(theta). There are thousands of lemmas in mathemat- ics, so in principle the same word play would work on any one of them, yet this joke based on Zorn’s lemma continues to be popular. Why should this 2There are several shaggy dog stories based on wordplay be so? The reason for this becomes clearer when in complex analysis. For example: we recall that the Axiom of Choice can be used to A bunch of Polish scientists decided to flee their re- prove many counterintuitive results in set theory, pressive government by hijacking an airliner and forcing such as the Banach-Tarski paradox. This leads the pilot to fly them to a western country. They drove to some mathematicians to reject the Axiom of Choice. the airport, forced their way on board a large passenger Yet because it is so useful (and because it seems jet, and found there was no pilot on board. Terrified, they listened as the sirens got louder. Finally, one of the sci- so innocuous at first glance), many mathemati- entists suggested that since he was an experimentalist, cians accept it, but with strong reservations. They he would try to fly the aircraft. He sat down at the con- find it distasteful, or unpalatable, or, if you will, a trols and tried to figure them out. The sirens got louder bitter fruit. and louder. Armed men surrounded the jet. The would- Here is another one, even more revealing: be pilot’s friends cried out, “Please, please take off now!!! Hurry!!!” The experimentalist calmly replied, “Have pa- Q: What is brown, furry, runs to the sea, and is equiv- tience. I’m just a simple pole in a complex plane.” alent to the Axiom of Choice? A group of Polish tourists is flying on a small airplane A: Zorn’s lemming. through the Grand Canyon on a sightseeing tour. The tour guide announces: “On the right of the airplane, you can see the famous Bright Angle Falls.” The tourists leap out Lemmings are known to follow each other of their seats and crowd to the windows on the right side. blindly into the sea, where they drown. The impli- This causes a dynamic imbalance, and the plane violently cation of this version seems to be that mathe- rolls to the side and crashes into the canyon wall. All maticians who rely upon the Axiom of Choice are aboard are lost. The moral of this episode is: always keep all lemmings who blindly follow one another and your poles off the right side of the plane. who may all be headed for intellectual death. For a classification of shaggy dog stories, including ones with punch lines based on “an Axiom of Science”, see [B].

JANUARY 2005 NOTICES OF THE AMS 25 Q: What do you get if you cross a mosquito with a Q: What is gray and huge and has integer coeffi- mountain climber? cients? A: You can’t cross a vector with a scalar. A: An elephantine equation. or this variation Q: What is very old, used by farmers, and obeys the Q: What do you get when you cross a mountain goat fundamental theorem of arithmetic? and a mountain climber? A: An antique tractorization domain. A: Nothing—you can’t cross two scalars.3 Q: What is hallucinogenic and exists for every group Q: What is a compact city? with order divisible by pk? A: It’s a city that can be guarded by finitely many A: A psilocybin p-subgroup. nearsighted policemen.4 Q: What is often used by Canadians to help solve cer- Q: What’s a dilemma? tain differential equations? A: A lemma that produces two results. A: The Lacross transform.

Q: What’s a polar bear? Q: What is clear and used by trendy sophisticated A: A rectangular bear after a coordinate transform. engineers to solve other differential equations? A: The Perrier transform. Q: What goes “Pieces of seven! Pieces of seven!” A: A parroty error. Q: Who knows everything there is to be known about vector analysis? Q: Why can’t you grow wheat in Z/6Z? A: The Oracle of del phi! A: It’s not a field. Q: Why can fish from the United States enter Cana- Q: What’s big, grey, and proves the uncountability dian waters without a passport? of the reals? A: This is permitted by the Law of Aquatic reci- A: Cantor’s diagonal elephant.5 procity!

3 Q: Why are topologists especially prone to malaria? For further discussion of cross breed , see [AH]. A: This disease comes from the Tietze fly!! 6 4Some mathematicians are so concerned about precision that they feel compelled to correct the mathematics in the Q: Why do truncated Maclaurin series fit the origi- jokes themselves. The following email comment on this joke nal function so well? appeared at http://www.xs4all.nl/~jcdverha/ A: Because they are “Taylor” made. scijokes/1_6.html: From: http://wheierman# NoSpam.corunduminium.com (Will Heierman): Re- cently, I read the following on a math Q: What is locally like a ring and very evil? joke website. It was attributed to Peter Lax. A: A devilish scheme. “What is a compact city?” Q: What is a proof? “A city that can be guarded by a finite number of near- sighted policemen.” A: One-half percent of alcohol. However, I doubt that he would make such a mis- Q: Can you prove Lagrange’s Identity? take. Moreover, he was my advisor when I was a graduate student, and I actually recall a con- A: Are you kidding? It’s really hard to prove the iden- versation with him regarding this anecdote. It tity of someone who’s been dead for over 150 years! did not go exactly like this (but this makes a better story): Q: What is black and white ivory and fills space? “Dr. Lax, wouldn’t it be better to say that a compact city A: A piano curve. is one that can be guarded by a finite number of police- men, no matter how nearsighted they are?” Q: What’s polite and works for the phone company? “That’s not any better, really, for if the nth policeman could A: A deferential operator. only see a distance of 1/2(n+2) , no finite number of them could guard even [0, 1]!” Q: What does an analytic number theorist say when “Wow! How do we handle this?” he is drowning? A: Log-log, log-log, log-log, …. “I might reword it slightly: A compact city is one which can be guarded by a finite number of policemen, no matter 6This joke contains an epidemiological error and should how nearsighted a policeman is.” more correctly read “Why are topologists especially prone 5For a discussion of the significance of other jokes in the to sleeping sickness?”, as it is sleeping sickness, not malaria, elephant cycle, see [AD]. that is spread by the Tsetse fly.

26 NOTICES OF THE AMS VOLUME 52, NUMBER 1 Q: What does a topologist call a virgin? following text shows that mathematicians are so A: Simply connected.7 clever that they can imagine a lightbulb that changes itself! Some joking questions are adaptations of stan- dard folkloristic forms. The question “How many Q: How many lightbulbs does it take to change a xxxx’s does it take to change a lightbulb” has sev- lightbulb? eral answers [D2], including the following: A: One, if it knows its own Gödel number.

Q: How many topologists does it take to change a This joke also expresses the fear that mathe- lightbulb? matics itself may be inconsistent (and perhaps A: Just one, but what will you do with the dough- even lead to paradoxes), as Gödel numbers were nut? used by Gödel to prove his two celebrated incom- pleteness theorems, both of which shook the foun- Q: How many number theorists does it take to dations of mathematics. The same anxiety sur- change a lightbulb? faces in the following parody of the traditional A: This is not known, but it is conjectured to be an joke: “Why did the chicken cross the road? To get elegant prime. to the other side.”

Q: How many geometers does it take to screw in a Q: Why did the chicken cross the road? lightbulb? A: Gödel: It cannot be proved whether the chicken A: None. You can’t do it with a straightedge and com- crossed the road. pass. Some other answers to this question provided Q: How many analysts does it take to screw in a light- by famous mathematicians include: bulb? A: Three. One to prove existence, one to prove Q: Why did the chicken cross the road? uniqueness, and one to derive a nonconstructive A: Erdös: It was forced to do so by the chicken-hole algorithm to do it. principle.

Q: How many Bourbakists does it take to replace a Q: Why did the chicken cross the road? lightbulb? A: Riemann: The answer appears in Dirichlet’s lec- A: Changing a lightbulb is a special case of a more tures. general theorem concerning the maintenance and repair of an electrical system. To establish upper and Q: Why did the chicken cross the road? lower bounds for the number of personnel required, A: Fermat: It did not fit on the margin on this side. we must determine whether the sufficient conditions of Lemma 2.1 (Availability of personnel) and those A good mathematician always looks for possi- of Corollary 2.3.55 (Motivation of personnel) apply. ble counterexamples to unproved conjectures. Here If and only if these conditions are met, we derive the is an amusing variant of the chicken crossing joke, result by an application of the theorems in Section demonstrating that the notion of “other side” is not 3.1123. The resulting upper bound is, of course, a always as straightforward as it seems: result in an abstract measure space, in the weak-* topology. Q: Why did the chicken cross the Möbius strip? A: To get to the other–er…. Q: How many mathematicians does it take to screw in a lightbulb? If you cannot prove the theorem, prove a dif- A: 0.999999…. ferent one:

In addition to making fun of beginning mathe- Q: Why did the chicken cross the Möbius strip? matics students who have difficulty grasping the A: To get to the same side. equality 0.9999 ···=1, this joke also shows how mathematics is essentially a solitary endeavor, as Although puns and one-liners occur orally, mod- it only takes one mathematician to do it. This can ern folklore is transmitted by photocopier, email, be contrasted with many of the other standard an- and the Internet, frequently in the form of lists [DP]. swers to this question, which demonstrate that An example of this form is the following “Top Ten groups of people are often required. In fact, the Excuses for Not Doing Homework”:

7Here, again, the jokester reveals a lack of understand- • I accidentally divided by zero and my paper burst ing of either mathematics or anatomy. into flames.

JANUARY 2005 NOTICES OF THE AMS 27 • Isaac Newton’s birthday. Proof by picture: • I could only get arbitrarily close to my textbook. A more convincing form of proof by example. Com- I couldn’t actually reach it. bines well with proof by omission. • I have the proof, but there isn’t room to write it in this margin. Proof by intimidation: “Trivial.” • I was watching the World Series and got tied up trying to prove that it converged. Proof by seduction: • I have a solar-powered calculator and it was “Convince yourself that this is true!” cloudy. • I locked the paper in my trunk, but a four- Proof by cumbersome notation: dimensional dog got in and ate it. Best done with access to at least four alphabets and • I couldn’t figure out whether i am the square of special symbols. negative one or i is the square root of negative one. Proof by exhaustion: • I took time out to snack on a doughnut and a cup An issue or two of a journal devoted to your proof of coffee [and] I spent the rest of the night try- is useful. ing to figure which one to dunk. • I could have sworn I put the home work inside a Proof by obfuscation: Klein bottle, but this morning I couldn’t find it. A long plotless sequence of true and/or meaning- less syntactically related statements. Every mathematician relies on the truth of prior propositions proved by other mathematicians, and Proof by wishful citation: it is ordinarily impossible for a person to check the The author cites the negation, converse, or gener- validity of every proof upon which his work is alization of a theorem from the literature to sup- based. For this reason, mathematicians place port his claims. tremendous emphasis on the correctness of mathematical proofs. The following list of proof Proof by eminent authority: techniques hints at the anxieties felt by many “I saw Karp in the elevator and he said it was prob- mathematicians regarding the degree to which ably NP-complete.” mathematical truth is dependent upon the trust- worthiness of previous results. This anxiety is ex- Proof by personal communication: acerbated by the fact that some mathematicians “Eight-dimensional colored cycle stripping is NP- have a less rigorous proof style than other math- complete [Karp, personal communication].” ematicians. How to prove it. Guide for lecturers. Proof by reduction to the wrong problem: “To see that infinite-dimensional colored cycle Proof by vigorous handwaving: stripping is decidable, we reduce it to the halting Works well in a classroom or seminar setting. problem.”

Proof by forward reference: Proof by reference to inaccessible literature: Reference is usually to a forthcoming paper of the The author cites a simple corollary of a theorem author, which is often not as forthcoming as at first. to be found in a privately circulated memoir of the Slovenian Philological Society, 1883. Proof by funding: How could three different government agencies Proof by importance: be wrong? A large body of useful consequences all follow from the proposition in question. Proof by example: The author gives only the case n =2and suggests Proof by accumulated evidence: that it contains most of the ideas of the general Long and diligent search has not revealed a coun- proof. terexample. Proof by omission: Proof by cosmology: “The reader may easily supply the details.” The negation of the proposition is unimaginable or “The other 253 cases are analogous.” meaningless. Popular for proofs of the existence of God. Proof by deferral: “We’ll prove this later in the course.”

28 NOTICES OF THE AMS VOLUME 52, NUMBER 1 Proof by mutual reference: work may be based upon false premises. There are In reference A, Theorem 5 is said to follow from many fallacious proofs of the type purporting to Theorem 3 in reference B, which is shown to fol- prove that 1=2. Here is just one example using the low from Corollary 6.2 in reference C, which is an ever-popular trick of dividing by zero. easy consequence of Theorem 5 in reference A. Theorem. 3=4. Proof by metaproof: Proof. Suppose A method is given to construct the desired proof. a + b = c. The correctness of the method is proved by any of This can also be written as: these techniques. 4a − 3a +4b − 3b =4c − 3c . After reorganizing: Proof by vehement assertion: 4a +4b − 4c =3a +3b − 3c. It is useful to have some kind of authority relation Take the constants out of the brackets: to the audience. 4(a + b − c)=3(a + b − c). Remove the same term left and right: Proof by ghost reference: 4=3. Nothing even remotely resembling the cited theo- rem appears in the reference given. The next joke originates from the time before calculators, when scientists used slide rules or ta- Proof by semantic shift: bles of logarithms to carry out complex calculations: Some of the standard but inconvenient definitions are changed for the statement of the result. You know that during the Great Flood, Noah brought along two of every species Proof by appeal to intuition: for reproductive purposes. Well, after a Cloud-shaped drawings frequently help here. few weeks on the ark, all the couples were getting along fine, except for these In common parlance, if you ask someone to two snakes. Day and night, Noah wor- prove something, it can be considered a challenge, ried that this was going to mean the end which might lead to conflict. Even among mathe- of this species. Finally when the flood maticians, there are those who insist on a high de- ended and the ark hit ground, the two gree of rigor and those who are more willing to ac- snakes darted out of the ship and headed cept less rigorous standards: to the nearest picnic table where they What is the difference between an ar- started to “go at it”. It was then that gument and a proof? An argument will Noah realized that…Adders can’t mul- convince a reasonable man, but a proof tiply without their log tables. is needed to convince an unreasonable Here is another version: one. The Flood is over and the ark has landed. There are also parodies of the logic utilized in Noah lets all the animals out and says, mathematical proofs: “Go forth and multiply.” A few months Theorem. A cat has nine tails. later, Noah decides to take a stroll and Proof. No cat has eight tails. A cat has one more see how the animals are doing. Every- tail than no cat. Therefore a cat has nine tails. where he looks he finds baby animals. Everyone is doing fine except for one and a variant: pair of little snakes. “What’s the prob- lem?” says Noah. “Cut down some trees Theorem. All dogs have nine legs. and let us live there,” say the snakes. Proof. Would you agree that no dog has five Noah follows their advice. Several more legs? Would you agree that a dog has four legs weeks pass. Noah checks on the snakes more than no dog? 4+5=? again. Lots of little snakes, everybody is Theorem. All positive integers are interesting. happy. Noah asks, “Want to tell me how Proof. Assume the contrary. Then there is a low- the trees helped?” “Certainly”, say the est noninteresting positive integer. But, hey, that’s snakes. “We’re adders, so we need logs pretty interesting! A contradiction. to multiply.” The following class of joke is often told by math- There are many mathematical limericks and ematicians when commenting on the common mis- songs, but in the interests of brevity we include only takes of nonmathematicians in attempting to con- the following. It is based upon a traditional song struct valid proofs. But these jokes are also a “One Hundred Bottles of Beer on the Wall” which reflection of the worry that perhaps one’s own could be written in mathematical notation:

JANUARY 2005 NOTICES OF THE AMS 29 N bottles of beer on the wall, so forth, extinguishes the fire with the N bottles of beer, minimum amount of water and energy You take one down, and pass it around, needed. Later the mathematician wakes N − 1 bottles of beer on the wall. up and smells smoke. He goes to the hall, sees the fire and then the fire hose. He The song begins at N = 100 and then repeats until thinks for a moment and then exclaims, there are no bottles of beer remaining. It is often “Ah, a solution exists!” and then goes sung by children on long car rides, as a way of keep- back to bed. ing them busy. The following mathematical variant is an amusing way to keep the children occupied Here is another version: a bit longer: A physicist and a mathematician are Aleph-null bottles of beer on the wall, sitting in a faculty lounge. Suddenly, the Aleph-null bottles of beer, coffee machine catches on fire. The You take one down, and pass it physicist grabs a bucket and leaps to- around, ward the sink, fills the bucket with water, Aleph-null bottles of beer on the wall. and puts out the fire. Second day, the same two sit in the same lounge. Again Making Fun of Mathematicians the coffee machine catches on fire. This Most of the above texts would not be likely to cir- time, the mathematician stands up, gets culate among nonmathematicians, as they rely on a bucket, and hands the bucket to the specialized knowledge of the field. However, there physicist, thus reducing the problem to is a substantial body of humor involving mathe- a previously solved one. maticians themselves, typically contrasted with The principle of reducing a theorem to a previ- other scientists, which is more accessible to a gen- ously proved theorem is often misinterpreted by eral audience [G]. These jokes are often told by nonmathematicians as that of reducing a “problem” physicists and engineers, but they are also related to a previously solved one. The result is some- by mathematicians, despite the fact that they are times a bit unexpected: typically the object of ridicule. One reason for this may be that the mathematician is more concerned A mathematician and an engineer are with ideas than with reality: on a desert island. They find two palm trees with one coconut each. The engi- Engineers think that equations approx- neer shinnies up one tree, gets the co- imate the real world. Physicists think conut, and eats it. The mathematician that the real world approximates equa- shinnies up the other tree, gets the co- tions. Mathematicians are unable to conut, climbs the other tree and puts it make the connection. there. “Now we’ve reduced it to a prob- Another variant of this reads: lem we know how to solve.” An engineer thinks that his equations are In the following story the mathematician is so an approximation to reality. A physicist wrapped up in his own mathematical world that he thinks reality is an approximation to his appears heartless: equations. A mathematician doesn’t There are a mathematician and a physi- care. cist and a burning building with people The stereotypical mathematician is depicted as inside. There are a fire hydrant and a being impractical, abstract, and not concerned hose on the sidewalk. The physicist has about everyday affairs: to put the fire out…so, he attaches the hose to the hydrant, puts the fire out, and An engineer, a physicist, and a mathe- saves the house and the family. Then matician are staying in a hotel. The en- they put the people back in the house, set gineer wakes up and smells smoke. He it on fire, and ask the mathematician goes out into the hallway and sees a fire, to solve the problem. So, he takes the so he fills a trash can from his room hose off the hydrant and lays it on the with water and douses the fire. He goes sidewalk. “Now I’ve reduced it to a back to bed. Later, the physicist wakes previously solved problem,” and walks up and smells the smoke. He opens his away. door and sees a fire in the hallway. He walks down the hall to a fire hose and The mathematical solution to a problem is often after calculating the flame velocity, dis- the most elegant but does not solve the original tance, water pressure, trajectory, and problem in an effective manner:

30 NOTICES OF THE AMS VOLUME 52, NUMBER 1 One day a farmer called up an engi- mad scientist went to the engineer’s cell neer, a physicist, and a mathematician and found it long empty. The engineer and asked them to fence in the largest had constructed a can opener from possible area with the least amount of pocket trash, used aluminum shavings fence. The engineer made the fence in and dried sugar to make an explosive, a circle and proclaimed that he had the and escaped. The physicist had worked most efficient design. The physicist made out the angle necessary to knock the lids a long, straight line and proclaimed “We off the tin cans by throwing them against can assume the length is infinite…” and the wall. She was developing a good pointed out that fencing off half of the pitching arm and a new quantum the- Earth was certainly a more efficient way ory. The mathematician had stacked the to do it. The mathematician just laughed unopened cans into a surprising solution at them. He built a tiny fence around to the kissing problem; his desiccated himself and said, “I declare myself to be corpse was propped calmly against a on the outside.” wall, and this was inscribed on the floor Some mathematical constructions, when applied in blood: Theorem: If I can’t open these to the real world, give rather absurd results: cans, I’ll die. Proof: Assume the oppo- site.… A biologist, a physicist, and a mathe- matician were sitting in a street café In the following story the mathematician ap- watching the crowd. Across the street plies flawless logic, but shows an unwillingness to they saw a man and a woman entering generalize from a single case, a procedure that a building. Ten minutes later they reap- holds many mathematical pitfalls, but which is peared together with a third person. often valid in the real world: “They have multiplied,” said the biologist. A mathematician, a physicist, and an “Oh no, an error in measurement,” the engineer were traveling through Scot- physicist sighed. “If exactly one person land when they saw a black sheep enters the building now, it will be empty through the window of the train. “Aha,” again,” the mathematician concluded. says the engineer, “I see that Scottish sheep are black.” “Hmm,” says the physi- Three men with degrees in mathematics, cist, “You mean that some Scottish sheep physics, and biology are locked up in dark are black.” “No,” says the mathemati- rooms for research reasons. A week later cian, “All we know is that there is at the researchers open the door and the bi- least one sheep in Scotland, and that at ologist steps out and reports: “Well, I sat least one side of that one sheep is black!” around until I started to get bored, then I searched the room and found a tin which Here, the mathematician again gives impracti- I smashed on the floor. There was food cal answers to real-world questions: in it which I ate when I got hungry. That’s Three men are in a hot-air balloon. Soon, it.” Then they free the man with the de- they find themselves lost in a canyon gree in physics and he says: “I walked somewhere. One of the three men says, along the walls to get an image of the “I’ve got an idea. We can call for help in room’s geometry, then I searched it. There this canyon and the echo will carry our was a metal cylinder at five feet into the voices far.” So he leans over the basket room and two feet left of the door. It felt and yells out, “Helloooooo! Where are like a tin and I threw it at the left wall at we?” (They hear the echo several times.) the right angle and velocity for it to crack Fifteen minutes later, they hear this echo- open.” Finally, the researchers open the ing voice: “Hellooooo! You’re lost!!” One third door and hear a faint voice out of of the men says, “That must have been the darkness: “Let C be an open can.” a mathematician.” Puzzled, one of the Here is a variant with a more dire consequence: other men asks, “Why do you say that?” The reply: “For three reasons: (1) He took There was a mad scientist (a mad … a long time to answer, (2) he was social…scientist) who kidnapped three absolutely correct, and (3) his answer colleagues, an engineer, a physicist, and was absolutely useless.” a mathematician, and locked each of them in separate cells with plenty of Although the three points made in the man’s canned food and water but no can reply do apply to many mathematicians, there is opener. A month later, returning, the more that is unspoken. After all, there is something

JANUARY 2005 NOTICES OF THE AMS 31 a bit strange about the mathematician saying that the interaction, the student is startled to hear the the ballooners are lost, for one would think that, professor ask, “Which way was I headed when we if the mathematician knew where he was, he would met?” The student points, saying, “You were going tell them. The implication is clearly that the math- that way, sir.” “Good,” says Wiener, “Then I’ve had my ematician is as lost as they are, wandering about lunch.” But probably the best-known Wiener absent- in a mathematical wilderness, perhaps never to re- minded-professor anecdote goes as follows: turn to civilization. Here is an interesting metajoke: One day the Wiener family was scheduled to move into a new house. Mrs. Wiener, An engineer, a physicist, and a mathe- mindful of her husband’s propensity for matician find themselves in an anec- forgetting, wrote the new address on a dote, indeed an anecdote quite similar to slip of paper and handed it to him. He many that you have no doubt already scoffed, saying, “I wouldn’t forget such heard. After some observations and an important thing,” but he took the slip rough calculations the engineer realizes of paper and put it in his pocket. Later the situation and starts laughing. A few that same day at the university a col- minutes later the physicist understands league came by his office with an inter- too and chuckles to himself happily, as esting problem. Wiener searched for a he now has enough experimental evi- piece of paper and took the slip from his dence to publish a paper. This leaves pocket to use to write some mathemati- the mathematician somewhat perplexed, cal equations. When he finished, he crum- as he had observed right away that he pled up the slip of paper and threw it was the subject of an anecdote and de- away. That evening, he remembered duced quite rapidly the presence of there was something about a new house humor from similar anecdotes, but con- but he couldn’t find the slip of paper siders this anecdote to be too trivial a with the address on it. Without any al- corollary to be significant, let alone ternative course of action, he returned to funny. his old home, where he spotted a little girl This metajoke says a lot about mathematicians. on the sidewalk. “Say, little girl,” he said, First, they are often very quick thinkers, able to “Do you know where the Wieners live?” reach conclusions far faster than others. Second, The girl replied, “That’s o.k., Daddy, they can see the humor in some jokes but are eas- Mommy sent me to get you.” ily bored by the routine or familiar. Third, they often There are apparently no limits on what an dismiss results that are obvious to themselves as absent-minded professor might forget, as attested “trivial”, even though the results may not be triv- ial to others. The following joke vividly illustrates by the following lesser-known Wiener anecdote [K]: this penchant. One day, a student saw Wiener in the A mathematics professor was lecturing post office and wanted to introduce him- to a class of students. As he wrote some- self to the famous professor. After all, thing on the board, he said to the class how many M.I.T. students could say that “Of course, this is immediately obvious.” they had actually shaken the hand of Upon seeing the blank stares of the stu- Norbert Wiener? However, the student dents, he turned back to contemplate wasn’t sure how to approach the man. what he had just written. He began to The problem was aggravated by the fact pace back and forth, deep in thought. that Wiener was pacing back and forth, After about 10 minutes, just as the si- deeply lost in thought. Were the student lence was beginning to become uncom- to interrupt Wiener, who knows what fortable, he brightened, turned to the profound idea might be lost? Still, the stu- class and said, “Yes, it IS obvious.” dent screwed up his courage and approached the great man. “Good morn- Some of the best-known examples of mathemat- ing, Professor Wiener,” he said. The pro- ical folk humor consist of anecdotes, often apoc- fessor looked up, struck his forehead, ryphal, attached to specific famous individuals. and said “That’s it: Wiener!” Not infrequently the protagonist is an absent-minded professor. In theory, it could be a professor from The stereotype of the absent-minded professor any academic discipline, but more often than not, it (mathematician) may contain a kernel of truth. The is a mathematician. One such classic text has Norbert degree of concentration required to solve a math- Wiener encountering a student on the street and en- ematical problem is such that one is virtually gaging him in theoretical discussion. At the end of obliged to put everything else aside for the moment

32 NOTICES OF THE AMS VOLUME 52, NUMBER 1 to devote all of one’s mental energies to working a calculator. At a morning press con- out a solution. ference, Attorney General John Ashcroft It is worth remarking that the corpus of math- said he believes the man is a member of ematical folk humor is not closed and that there the notorious al-gebra movement.9 He is are constant additions, often reflecting new trends being charged by the FBI with carrying and topical current events. Feminism may be partly weapons of math instruction. responsible for the following traditional joke that critiques the stereotyped notion that women are “Al-gebra is a fearsome cult,” Ashcroft unable to understand advanced mathematics (and said. “They desire average solutions by illustrates the stereotype of the arrogant mathe- means and extremes, and sometimes go matician):8 off on tangents in a search of absolute value. They use secret code names like Two mathematicians are in a bar. The ‘x’ and ‘y’ and refer to themselves as first one says to the second that the av- ‘unknowns’, but we have determined erage person knows very little about they belong to a common denominator basic mathematics. The second one dis- of the axis with coordinates in every agrees and claims that most people can country. As the Greek philanderer Isosce- cope with a reasonable amount of math. les used to say, there are three sides to The first mathematician goes off to the every triangle,” Ashcroft declared. washroom, and in his absence the sec- ond calls over the waitress. He tells her When asked to comment on the arrest, that in a few minutes, after his friend has President Bush said, “If God had wanted returned, he will call her over and ask us to have better weapons of math in- her a question. All she has to do is an- struction, He would have given us more swer “one third x cubed.” She repeats fingers and toes. I am gratified that our “one thir–dex cue?” He repeats “one third government has given us a sine that it x cubed.” She asks, “one thir dex cuebd?” is intent on protracting us from those “Yes, that’s right,” he says. So she agrees, who are willing to disintegrate us with and goes off mumbling to herself, “one calculus disregard. Under the circum- thir dex cuebd…”. The first guy returns ferences, we must differentiate their and the second proposes a bet to prove root, make our point, and draw the line.” his point, that most people do know President Bush warned, “These weapons something about basic math. He says of math instruction have the potential he will ask the blonde waitress an inte- to decimal everything in their math on gral, and the first laughingly agrees. a scalene never before seen unless we The second man calls over the waitress become exponents of a higher power and asks “what is the integral of x and begin to factor in random facts of squared?” The waitress says “one third vertex.” x cubed” and while walking away, turns back and says over her shoulder, “plus Attorney General Ashcroft said, “Read a constant!” my ellipse. Their days are numbered as The following story illustrates the influence of the hypotenuse tightens around their contemporary events on mathematical humor. As necks.” with many jokes, it reflects the fears and anxieties of the general public. Indeed, the joke goes so far What Is Real? as to equate mathematics teachers with terrorists The abundance of joking questions involving plays (recall the Unabomber, Theodore Kaczynski), which on mathematical terminology, as well as the jokes perhaps explains why mathematics often fails to involving the stereotype of the mathematician, sug- get the support it deserves! gest that mathematicians like to play. The delight At New York’s Kennedy Airport today, an in playfulness would seem to run counter to the individual was arrested trying to board stereotype of the humorless pedant who is con- a flight while in possession of a ruler, a cerned only with precision, but there is no neces- protractor, a setsquare, a slide rule, and sary contradiction. A predilection for fantasy and

8It is also interesting that, unlike the traditional blonde jokes, in which the blonde is depicted as being stupid, in 9Of course, one irony in this joke is that the word algebra this joke the blonde outsmarts the mathematician. For more derives from the word al-Jabr that appears in the title of on the cycle of blonde jokes, see [DP] and the references a book written in Baghdad around 825 by the Arab math- therein. ematician Al-Khwarizmi.

JANUARY 2005 NOTICES OF THE AMS 33 whimsy is perhaps related to the ability of the [DP] A. DUNDES and C. PAGTER, Why Don’t Sheep Shrink When mathematician to invent new mathematical con- it Rains?, Syracuse University Press, Syracuse, 2000. cepts. But it also leads to the stereotype of the [G] C. GILKEY, The physicist, the mathematician and the mathematician as inhabiting his own mathemati- engineer: Scientists and the professional slur, Western Folklore 49 (1990), 215–220. cal universe, to the point where he is unable to cope [J] B. JACKSON, The greatest mathematician in the world: with everyday problems such as putting out a fire Norbert Wiener stories, Western Folklore 31 (1972), or eating a coconut. In fact, the mathematician is 1–22. often lost, both literally and metaphorically. [K] S. KRANTZ, Mathematical anecdotes, The Mathematical As for the disparity between the mathematical Intelligencer 12 (4) (1990), 32–38. world and the real world, one joke text claims that the mathematician is unaware of it, while another claims he is aware of it but does not care. The im- plication is that many mathematicians feel much more at home in the mathematical universe than in the mundane, quotidian, contingent world. The goal of all mathematics is to discover mathemati- cal truth, which does not depend on the here and now. The difficulty, though, is that mathematics cannot appeal to the logic of reality to validate its axiomatic foundation. This is why there is such anx- iety about the consistency of axiomatic systems and the validity of mathematical proofs, as they are the only means of achieving mathematical certainty. Mathematical jokes allude to and confirm and validate the existence of the mathematical uni- verse, and they can only be understood and ap- preciated by the mathematical cognoscenti. The puns reveal what happens when the two worlds col- lide, and pure mathematics becomes tainted or corrupted or trivialized by its encounter. The home- work jokes reveal how difficult it is to inhabit both worlds, as many truths in one world do not trans- late into truths in the other. Yet mathematics is clearly a human endeavor and can solve concrete problems. It is precisely the tension between the mathematical universe and the nonmathematical universe that is central to much of mathematical humor.

Acknowledgements One of us [PR] would like to thank his wife, Alison Dundes Renteln, for her contributions to this arti- cle and for putting up with his often incompre- hensible jokes.

References [AD] R. ABRAHAMS and A. DUNDES, On elephantasy and ele- phanticide, The Psychoanalytic Review 56 (1969), 225–241. [AH] R. D. ABRAHAMS and J. C. HICKERSON, Cross-fertiliza- tion riddles, Western Folklore 23 (1964), 253–257. [B] J. BRUNVAND, A classification for shaggy dog stories, Journal of American Folkore 76 (1963), 42–68. [D1] A. DUNDES ed., International Folkloristics: Classic Con- tributions by the Founders of Folklore, Rowman and Lit- tlefield Publishers, Lanham, 1999. [D2] ——— , Many hands make light work or caught in the act of screwing in lightbulbs, Western Folklore 40 (1981), 261–266.

34 NOTICES OF THE AMS VOLUME 52, NUMBER 1