My 2018 Mathematics A To Z: Jokes


For today’s entry, Iva Sallay, of Find The Factors, gave me an irresistible topic. I did not resist.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Jokes.

What’s purple and commutes?
An Abelian grape.

Whatever else you say about mathematics we are human. We tell jokes. I will tell some here. You may not understand the words in them. That’s all right. From the Abelian grape there, you gather this is some manner of wordplay. A pun, particularly. It’s built on a technical term. “Abelian groups” come from (not high school) Algebra. In an Abelian group, the group multiplication commutes. That is, if ‘a’ and ‘b’ are any things in the group, then their product “ab” is the same as “ba’. That is, the group works like ordinary addition on numbers does. We say “Abelian” in honor of Niels Henrik Abel, who taught us some fascinating stuff about polynomials. Puns are a common kind of humor. So common, they’re almost base. Even a good pun earns less laughter than groans.

But mathematicians make many puns. A typical page of mathematics jokes has a whole section of puns. “What’s yellow and equivalent to the Axiom of Choice? Zorn’s Lemon.” “What’s nonorientable and lives in the sea?” “Möbius Dick.” “One day Jesus said to his disciples, `The Kingdom of Heaven is like 3x2 + 8x – 9′. Thomas looked very confused and asked peter, `What does the teacher mean?’ Peter replied, `Don’t worry. It’s just another one of his parabolas’.” And there are many jokes built on how it is impossible to tell the difference between the sounds of “π” and “pie”.

It shouldn’t surprise that mathematicians make so many puns. Mathematics trains people to know definitions. To think about precisely what we mean. Puns ignore definitions. They build nonsense out of the ways that sounds interact. Mathematicians practice how to make things interact, even if they don’t know or care what the underlying things are. If you’ve gotten used to proving things about aba^{-1}b^{-1} , without knowing what ‘a’ or ‘b’ are, it’s difficult to avoid turning “poles on the half-plane” (which matters in some mathematical physics) to a story about Polish people on an aircraft.

Popeye's lousy tutor: 'Today I am going to test you at mental multiplication. Quick, how much is 6 1/2 times 656? Quick!' Popeye: '4,264.' 'Right!' 'Blow me down! Anybody what can guess like that don't need no edjacation!'
Elzie Segar’s Thimble Theater from the 14th of September, 1929. Rerun on ComicsKingdom on the 26th of February, 2016. That’s Bernice, the magical Whiffle Hen, as the strange birdlike creature in the last panel there.

If there’s a flaw to this kind of humor it’s that these jokes may sound juvenile. One of the first things that strikes kids as funny is that a thing might have several meanings. Or might sound like another thing. “Why do mathematicians like parks? Because of all the natural logs!”

Jokes can be built tightly around definitions. “What do you get if you cross a mosquito with a mountain climber? Nothing; you can’t cross a vector with a scalar.” “There are 10 kinds of people in the world, those who understand binary mathematics and those who don’t.” “Life is complex; it has real and imaginary parts.”

Paige: 'I keep forgetting ... what's the cosine of 60 degrees?' Jason: 'Well, let's see. If I recall correctly ... 1 - (pi/3)^2/2! + (pi/3)^4/4! - (pi/3)^6/6! + (pi/3)^8/8! - (pi/3)^10/10! + (pi/3)^12/12! - (and this goes on a while, up to (pi/3)^32/32! - ... )' Paige: 'In case you've forgotten, I'm not paying you by the hour.' Jason: '1/2'.
Bill Amend’s FoxTrot Classics for the 23rd of May, 2018. It originally ran the 29th of May, 1996.

There are more sophisticated jokes. Many of them are self-deprecating. “A mathematician is a device for turning coffee into theorems.” “An introvert mathematician looks at her shoes while talking to you. An extrovert mathematician looks at your shoes.” “A mathematics professor is someone who talks in someone else’s sleep”. “Two people are adrift in a hot air balloon. Finally they see someone and shout down, `Where are we?’ The person looks up, and studies them, watching the balloon drift away. Finally, when they are barely in shouting range, the person on the ground shouts back, `You are in a balloon!’ The first passenger curses their luck at running across a mathematician. `How do you know that was a mathematician?’ `Because her answer took a long time, was perfectly correct, and absolutely useless!”’ These have the form of being about mathematicians. But they’re not really. It would be the same joke to say “a poet is a device for turning coffee into couplets”, the sleep-talker anyone who teachers, or have the hot-air balloonists discover a lawyer or a consultant.

Some of these jokes get more specific, with mathematics harder to extract from the story. The tale of the nervous flyer who, before going to the conference, sends a postcard that she has a proof of the Riemann hypothesis. She arrives and admits she has no such thing, of course. But she sends that word ahead of every conference. She knows if she died in a plane crash after that, she’d be famous forever, and God would never give her that. (I wonder if Ian Randal Strock’s little joke of a story about Pierre de Fermat was an adaptation of this joke.) You could recast the joke for physicists uniting gravity and quantum mechanics. But I can’t imagine a way to make this joke about an ISO 9000 consultant.

'If it's a hunnert miles to th' city an' a train is travelin' thurty miles an hour is due t'arrive at 5:00 pm --- what time does th' train leave Hootin' Holler, Jughaid?' 'I dunno, Miz Prunelly, but you better go now jest t'be on th' safe side!!'
John Rose’s Barney Google and Snuffy Smith for the 12th of February, 2016.

A dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”

This joke is very mathematical. I know of no important results actually based on spherical cows. But the attitude that tries to make spheres of cows comes from observing mathematicians. To describe any real-world process is to make a model of that thing. A model is a simplification of the real thing. You suppose that things behave more predictably than the real thing. You trust the error made by this supposition is small enough for your needs. A cow is complicated, all those pointy ends and weird contours. A sphere is easy. And, besides, cows are funny. “Spherical cow” is a funny string of sounds, at least in English.

The spherical cows approach parodying the work mathematicians do. Many mathematical jokes are burlesques of deductive logic. Or not even burlesques. Charles Dodgson, known to humans as Lewis Carroll, wrote this in Symbolic Logic:

“No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
∴ This party of tourists need not run.”

[ Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.

He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premises to be true, I should be quite clear that I needn’t run — and I should walk!

And you will reply “But suppose there was a mad bull behind you?”

And then your innocent friend will say “Hum! Ha! I must think that over a bit!” ]

The punch line is diffused by the text being so educational. And by being written in the 19th century, when it was bad form to excise any word from any writing. But you can recognize the joke, and why it should be a joke.

Not every mathematical-reasoning joke features some manner of cattle. Some are legitimate:

Claim. There are no uninteresting whole numbers.
Proof. Suppose there is a smalled uninteresting whole number. Call it N. That N is uninteresting is an interesting fact. Therefore N is not an uninteresting whole number.

Three mathematicians step up to the bar. The bartender asks, “you all want a beer?” The first mathematician says, “I don’t know.” The second mathematician says, “I don’t know.” The third says, “Yes”.

Some mock reasoning uses nonsense methods to get a true conclusion. It’s the fun of watching Mister Magoo walk unharmed through a construction site to find the department store exchange counter:

5095 / 1019 = 5095 / 1019 = 505 / 101 = 55 / 11 = 5

This one includes the thrill of division by zero.

The Venn Diagram of Grocery Shopping. Overlap 'have teenagers', 'haven't grocery shopped in two weeks', and 'grocery shopping on an empty stomach' and you get 'will need to go back in two days', 'bought entire bakery aisle', and 'bought two of everything'. Where they all overlap, 'need to take out second mortgage'.
Terri Libenson’s Pajama Diaries for the 16th of November, 2016. I was never one for buying too much of the bakery aisle, myself, but then I also haven’t got teenagers. And I did go through so much of my life figuring there was no reason I shouldn’t eat another bagel again.

Venn Diagrams are not by themselves jokes (most of the time). But they are a great structure for jokes. And easy to draw, which is great for us who want to be funny but don’t feel sure about their drafting abilities.

And then there are personality jokes. Mathematics encourages people to think obsessively. Obsessive people are often funny people. Alexander Grothendieck was one of the candidates for “greatest 20th century mathematician”. His reputation is that he worked so well on abstract problems that he was incompetent at practical ones. The story goes that he was demonstrating something about prime numbers and his audience begged him to speak about a specific number, that they could follow an example. And that he grumbled a bit and, finally, said, “57”. It’s not a prime number. But if you speak of “Grothendieck’s prime”, many will recognize what you mean, and grin.

There are more outstanding, preposterous personalities. Paul Erdös was prolific, and a restless traveller. The stories go that he would show up at some poor mathematician’s door and stay with them several months. And then co-author a paper with the elevator operator. (Erdös is also credited as the originator of the “coffee into theorems” quip above.) John von Neumann was supposedly presented with this problem:

Two trains are on the same track, 60 miles apart, heading toward each other, each travelling 30 miles per hour. A fly travels 60 miles per hour, leaving one engine flying toward the other. When it reaches the other engine it turns around immediately and flies back to the other engine. This is repeated until the two trains crash. How far does the fly travel before the crash?

The first, hard way to do this is to realize how far the fly travels is a series. The fly starts at, let’s say, the left engine and flies to the right. Add to that the distance from the right to the left train now. Then left to the right again. Right to left. This is a bunch of calculations. Most people give up on that and realize the problem is easier. The trains will crash in one hour. The fly travels 60 miles per hour for an hour. It’ll fly 60 miles total. John von Neumann, say witnesses, had the answer instantly. He recognized the trick? “I summed the series.”

Henry is frustrated with his arithmetic, until he goes to the pool hall and counts off numbers on those score chips.
Don Trachte’s Henry for the 6th of September, 2015.

The personalities can be known more remotely, from a handful of facts about who they were or what they did. “Cantor did it diagonally.” Georg Cantor is famous for great thinking about infinitely large sets. His “diagonal proof” shows the set of real numbers must be larger than the set of rational numbers. “Fermat tried to do it in the margin but couldn’t fit it in.” “Galois did it on the night before.” (Évariste Galois wrote out important pieces of group theory the night before a duel. It went badly for him. French politics of the 1830s.) Every field has its celebrities. Mathematicians learn just enough about theirs to know a couple of jokes.

Anthropomorphic 3/5: 'Honey, what's wrong?' Anthropomorphic 1/4: 'Sour son is leaving the faith! He said he's converting to decimals!'
Scott Hilburn’s The Argyle Sweater for the 9th of May, 2018. I like the shout-out to Archimedes in the background art, too. Archimedes, though, didn’t use fractions in the way we’d recognize them. He’d write out a number as a combination of ratios of some reference number. So he might estimate the length of something being as to the length of something else as 19 is to 7, or something like that. This seems like a longwinded and cumbersome way to write out numbers, or much of anything, and makes one appreciate his indefatigability as much as his insight.

The jokes can attach to a generic mathematician personality. “How can you possibly visualize something that happens in a 12-dimensional space?” “Easy, first visualize it in an N-dimensional space, and then let N go to 12.” Three statisticians go hunting. They spot a deer. One shoots, missing it on the left. The second shoots, missing it on the right. The third leaps up, shouting, “We’ve hit it!” An engineer and a mathematician are sleeping in a hotel room when the fire alarm goes off. The engineer ties the bedsheets into a rope and shimmies out of the room. The mathematician looks at this, unties the bedsheets, sets them back on the bed, declares, “this is a problem already solved” and goes back to sleep. (Engineers and mathematicians pair up a lot in mathematics jokes. I assume in engineering jokes too, but that the engineers make wrong assumptions about who the joke is on. If there’s a third person in the party, she’s a physicist.)

Do I have a favorite mathematics joke? I suppose I must. There are jokes I like better than others, and there are — I assume — finitely many different mathematics jokes. So I must have a favorite. What is it? I don’t know. It must vary with the day and my mood and the last thing I thought about. I know a bit of doggerel keeps popping into my head, unbidden. Let me close by giving it to you.

Integral z-squared dz
From 1 to the cube root of 3
   Times the cosine
   Of three π over nine
Equals log of the cube root of e.

This may not strike you as very funny. I’m not sure it strikes me as very funny. But it keeps showing up, all the time. That has to add up.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. Also, now and then, I talk about comic strips here. You might like that too.

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My 2018 Mathematics A To Z: Infinite Monkey Theorem


Dina Yagodich gave me the topic for today. She keeps up a YouTube channel with a variety of interesting videos. And she did me a favor. I’ve been thinking a long while to write a major post about this theorem. Its subject turns up so often. I’d wanted to have a good essay about it. I hope this might be one.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Infinite Monkey Theorem.

Some mathematics escapes mathematicians and joins culture. This is one such. The monkeys are part of why. They’re funny and intelligent and sad and stupid and deft and clumsy, and they can sit at a keyboard almost look in place. They’re so like humans, except that we empathize with them. To imagine lots of monkeys, and putting them to some silly task, is compelling.

Monkey Typewriter Theory: An immortal monkey pounding on a typewriter will eventually reproduce the text of 'Hamlet'. Baby Keyboard Theory: Left alone, a baby pounding on a computer keyboard will eventually order 32 cases of bathroom caulk from an online retailer.
Paul Trapp’s Thatababy for the 13th of February, 2014.

The metaphor traces back to a 1913 article by the mathematical physicist Émile Borel which I have not read. Searching the web I find much more comment about it than I find links to a translation of the text. And only one copy of the original, in French. And that page wants €10 for it. So I can tell you what everybody says was in Borel’s original text, but can’t verify it. The paper’s title is “Statistical Mechanics and Irreversibility”. From this I surmise that Borel discussed one of the great paradoxes of statistical mechanics. If we open a bottle of one gas in an airtight room, it disperses through the room. Why doesn’t every molecule of gas just happen, by chance, to end up back where it started? It does seem that if we waited long enough, it should. It’s unlikely it would happen on any one day, but give it enough days …

But let me turn to many web sites that are surely not all copying Wikipedia on this. Borel asked us to imagine a million monkeys typing ten hours a day. He posited it was possible but extremely unlikely that they would exactly replicate all the books of the richest libraries of the world. But that would be more likely than the atmosphere in a room un-mixing like that. Fair enough, but we’re not listening anymore. We’re thinking of monkeys. Borel’s is a fantastic image. It would see some adaptation in the years. Physicist Arthur Eddington, in 1928, made it an army of monkeys, with their goal being the writing all the books in the British Museum. By 1960 Bob Newhart had an infinite number of monkeys and typewriters, and a goal of all the great books. Stating the premise gets a laugh I doubt the setup would today. I’m curious whether Newhart brought the idea to the mass audience. (Google NGrams for “monkeys at typewriters” suggest that phrase was unwritten, in books, before about 1965.) We may owe Bob Newhart thanks for a lot of monkeys-at-typewriters jokes.

Kid: 'Mom, Dad, I want to go bungee jumping this summer!' Dad: 'A thousand monkeys working a thousand typewriters would have a better chance of randomly typing the complete works of William Shakespeare over the summer than you have of bungee jumping.' (Awksard pause.) Kid: 'What's a typewriter?' Dad: 'A thousand monkeys randomly TEXTING!'
Bill Hinds’s Cleats rerun for the 1st of July, 2018.

Newhart has a monkey hit on a line from Hamlet. I don’t know if it was Newhart that set the monkeys after Shakespeare particularly, rather than some other great work of writing. Shakespeare does seem to be the most common goal now. Sometimes the number of monkeys diminishes, to a thousand or even to one. Some people move the monkeys off of typewriters and onto computers. Some take the cowardly measure of putting the monkeys at “keyboards”. The word is ambiguous enough to allow for typewriters, computers, and maybe a Megenthaler Linotype. The monkeys now work 24 hours a day. This will be a comment someday about how bad we allowed pre-revolutionary capitalism to get.

The cultural legacy of monkeys-at-keyboards might well itself be infinite. It turns up in comic strips every few weeks at least. Television shows, usually writing for a comic beat, mention it. Computer nerds doing humor can’t resist the idea. Here’s a video of a 1979 Apple ][ program titled THE INFINITE NO. OF MONKEYS, which used this idea to show programming tricks. And it’s a great philosophical test case. If a random process puts together a play we find interesting, has it created art? No deliberate process creates a sunset, but we can find in it beauty and meaning. Why not words? There’s likely a book to write about the infinite monkeys in pop culture. Though the quotations of original materials would start to blend together.

But the big question. Have the monkeys got a chance? In a break from every probability question ever, the answer is: it depends on what the question precisely is. Occasional real-world experiments-cum-art-projects suggest that actual monkeys are worse typists than you’d think. They do more of bashing the keys with a stone before urinating on it, a reminder of how slight is the difference between humans and our fellow primates. So we turn to abstract monkeys who behave more predictably, and run experiments that need no ethical oversight.

Toby: 'So this English writer is like a genius, right? And he's the greatest playwright ever. And I want to be just like him! Cause what he does, see, is he gets infinite monkeys on typewriters and just lets 'em go nuts, so eventually they write ALL of Shakespeare's plays!' Brother: 'Cool! And what kind of monkey is an 'infinite'?' Toby: 'Beats me, but I hope I don't have to buy many of them.' Dad: 'Toby, are you *sure* ywou completely pay attention when your teachers are talking?' Toby: 'What? Yes! Why?'

Greg Cravens’ The Buckets for the 30th of March, 2014.

So we must think what we mean by Shakespeare’s Plays. Arguably the play is a specific performance of actors in a set venue doing things. This is a bit much to expect of even a skilled abstract monkey. So let us switch to the book of a play. This has a more clear representation. It’s a string of characters. Mostly letters, some punctuation. Good chance there’s numerals in there. It’s probably a lot of characters. So the text to match is some specific, long string of characters in a particular order.

And what do we mean by a monkey at the keyboard? Well, we mean some process that picks characters randomly from the allowed set. When I see something is picked “randomly” I want to know what the distribution rule is. Like, are Q’s exactly as probable as E’s? As &’s? As %’s? How likely it is a particular string will get typed is easiest to answer if we suppose a “uniform” distribution. This means that every character is equally likely. We can quibble about capital and lowercase letters. My sense is most people frame the problem supposing case-insensitivity. That the monkey is doing fine to type “whaT beArD weRe i BEsT tO pLAy It iN?”. Or we could set the monkey at an old typesetter’s station, with separate keys for capital and lowercase letters. Some will even forgive the monkeys punctuating terribly. Make your choices. It affects the numbers, but not the point.

Literary Calendar. Several jokes, including: Saturday 7pm: an infinite number of chimpanzees discuss their multi-volume 'Treasury of Western Literature with no Typos' at the Museum of Natural History. Nit picking to follow.
Richard Thompson’s Richard’s Poor Almanac rerun for the 7th of November, 2016.

I’ll suppose there are 91 characters to pick from, as a Linotype keyboard had. So the monkey has capitals and lowercase and common punctuation to get right. Let your monkey pick one character. What is the chance it hit the first character of one of Shakespeare’s plays? Well, the chance is 1 in 91 that you’ve hit the first character of one specific play. There’s several dozen plays your monkey might be typing, though. I bet some of them even start with the same character, so giving an exact answer is tedious. If all we want monkey-typed Shakespeare plays, we’re being fussy if we want The Tempest typed up first and Cymbeline last. If we want a more tractable problem, it’s easier to insist on a set order.

So suppose we do have a set order. Then there’s a one-in-91 chance the first character matches the first character of the desired text. A one-in-91 chance the second character typed matches the second character of the desired text. A one-in-91 chance the third character typed matches the third character of the desired text. And so on, for the whole length of the play’s text. Getting one character right doesn’t make it more or less likely the next one is right. So the chance of getting a whole play correct is \frac{1}{91} raised to the power of however many characters are in the first script. Call it 800,000 for argument’s sake. More characters, if you put two spaces between sentences. The prospects of getting this all correct is … dismal.

I mean, there’s some cause for hope. Spelling was much less fixed in Shakespeare’s time. There are acceptable variations for many of his words. It’d be silly to rule out a possible script that (say) wrote “look’d” or “look’t”, rather than “looked”. Still, that’s a slender thread.

Proverb Busters: testing the validity of old sayings. Doctor: 'A hundred monkeys at a hundred typewriters. Over time, will one of them eventually write a Shakepeare play?' Winky: 'Nope. Just the script for Grown-Ups 3'. Doctor: 'Another proverb busted.'
Tim Rickard’s Brewster Rockit for the 1st of April, 2014.

But there is more reason to hope. Chances are the first monkey will botch the first character. But what if they get the first character of the text right on the second character struck? Or on the third character struck? It’s all right if there’s some garbage before the text comes up. Many writers have trouble starting and build from a first paragraph meant to be thrown away. After every wrong letter is a new chance to type the perfect thing, reassurance for us all.

Since the monkey does type, hypothetically, forever … well, so each character has a probability of only \left(\frac{1}{91}\right)^{800,000} (or whatever) of starting the lucky sequence. The monkey will have 91^{800,000} chances to start. More chances than that.

And we don’t have only one monkey. We have a thousand monkeys. At least. A million monkeys. Maybe infinitely many monkeys. Each one, we trust, is working independently, owing to the monkeys’ strong sense of academic integrity. There are 91^{800,000} monkeys working on the project. And more than that. Each one takes their chance.

Melvin: 'Hold on now --- replacement? Who could you find to do all the tasks only Melvin can perform?' Rita: 'A macaque, in fact. Listen, if an infinite number of monkeys can write all the great works, I'm confident that one will more than cover for you.'
John Zakour and Scott Roberts’s Working Daze for the 29th of May, 2018.

There are dizzying possibilities here. There’s the chance some monkey will get it all exactly right first time out. More. Think of a row of monkeys. What’s the chance the first thing the first monkey in the row types is the first character of the play? What’s the chance the first thing the second monkey in the row types is the second character of the play? The chance the first thing the third monkey in the row types is the third character in the play? What’s the chance a long enough row of monkeys happen to hit the right buttons so the whole play appears in one massive simultaneous stroke of the keys? Not any worse than the chance your one monkey will type this all out. Monkeys at keyboards are ergodic. It’s as good to have a few monkeys working a long while as to have many monkeys working a short while. The Mythical Man-Month is, for this project, mistaken.

That solves it then, doesn’t it? A monkey, or a team of monkeys, has a nonzero probability of typing out all Shakespeare’s plays. Or the works of Dickens. Or of Jorge Luis Borges. Whatever you like. Given infinitely many chances at it, they will, someday, succeed.

Except.

A thousand monkeys at a thousand typewriters ... will eventually write 'Hamlet'. A thousand cats at a thousand typewriters ... will tell you go to write your own danged 'Hamlet'.
Doug Savage’s Savage Chickens for the 14th of August, 2018.

What is the chance that the monkeys screw up? They get the works of Shakespeare just right, but for a flaw. The monkeys’ Midsummer Night’s Dream insists on having the fearsome lion played by “Smaug the joiner” instead. This would send the play-within-the-play in novel directions. The result, though interesting, would not be Shakespeare. There’s a nonzero chance they’ll write the play that way. And so, given infinitely many chances, they will.

What’s the chance that they always will? That they just miss every single chance to write “Snug”. It comes out “Smaug” every time?

Eddie: 'You know the old saying about putting an infinite number of monkeys at an infinite number of typewriters, and eventually they'll accidentally write Shakespeare's plays?' Toby: 'I guess.' Eddie: 'My English teacher says that nothing about our class should worry those monkeys ONE BIT!'
Greg Cravens’s The Buckets for the 6th of October, 2018.

We can say. Call the probability that they make this Snug-to-Smaug typo any given time p . That’s a number from 0 to 1. 0 corresponds to not making this mistake; 1 to certainly making it. The chance they get it right is 1 - p . The chance they make this mistake twice is smaller than p . The chance that they get it right at least once in two tries is closer to 1 than 1 - p is. The chance that, given three tries, they make the mistake every time is even smaller still. The chance that they get it right at least once is even closer to 1.

You see where this is going. Every extra try makes the chance they got it wrong every time smaller. Every extra try makes the chance they get it right at least once bigger. And now we can let some analysis come into play.

So give me a positive number. I don’t know your number, so I’ll call it ε. It’s how unlikely you want something to be before you say it won’t happen. Whatever your ε was, I can give you a number M . If the monkeys have taken more than M tries, the chance they get it wrong every single time is smaller than your ε. The chance they get it right at least once is bigger than 1 – ε. Let the monkeys have infinitely many tries. The chance the monkey gets it wrong every single time is smaller than any positive number. So the chance the monkey gets it wrong every single time is zero. It … can’t happen, right? The chance they get it right at least once is closer to 1 than to any other number. So it must be 1. So it must be certain. Right?

Poncho, the dog, looking over his owner's laptop: 'They say if you let an infinite number of cats walk on an infinite number of keyboards, they'll eventually type all the great works of Shakespeare.' The cat walks across the laptop, connecting to their owner's bank site and entering the correct password. Poncho: 'I'll take it.'
Paul Gilligan’s Pooch Cafe for the 17th of September, 2018.

But let me give you this. Detach a monkey from typewriter duty. This one has a coin to toss. It tosses fairly, with the coin having a 50% chance of coming up tails and 50% chance of coming up heads each time. The monkey tosses the coin infinitely many times. What is the chance the coin comes up tails every single one of these infinitely many times? The chance is zero, obviously. At least you can show the chance is smaller than any positive number. So, zero.

Yet … what power enforces that? What forces the monkey to eventually have a coin come up heads? It’s … nothing. Each toss is a fair toss. Each toss is independent of its predecessors. But there is no force that causes the monkey, after a hundred million billion trillion tosses of “tails”, to then toss “heads”. It’s the gambler’s fallacy to think there is one. The hundred million billion trillionth-plus-one toss is as likely to come up tails as the first toss is. It’s impossible that the monkey should toss tails infinitely many times. But there’s no reason it can’t happen. It’s also impossible that the monkeys still on the typewriters should get Shakespeare wrong every single time. But there’s no reason that can’t happen.

It’s unsettling. Well, probability is unsettling. If you don’t find it disturbing you haven’t thought long enough about it. Infinities, too, are unsettling so.

Researcher overseeing a room of monkeys: 'Shakespeare would be OK, but I'd prefer they come up with a good research grant proposal.'
John Deering’s Strange Brew for the 20th of February, 2014.

Formally, mathematicians interpret this — if not explain it — by saying the set of things that can happen is a “probability space”. The likelihood of something happening is what fraction of the probability space matches something happening. (I’m skipping a lot of background to say something that simple. Do not use this at your thesis defense without that background.) This sort of “impossible” event has “measure zero”. So its probability of happening is zero. Measure turns up in analysis, in understanding how calculus works. It complicates a bunch of otherwise-obvious ideas about continuity and stuff. It turns out to apply to probability questions too. Imagine the space of all the things that could possibly happen as being the real number line. Pick one number from that number line. What is the chance you have picked exactly the number -24.11390550338228506633488? I’ll go ahead and say you didn’t. It’s not that you couldn’t. It’s not impossible. It’s just that the chance that this happened, out of the infinity of possible outcomes, is zero.

The infinite monkeys give us this strange set of affairs. Some things have a probability of zero of happening, which does not rule out that they can. Some things have a probability of one of happening, which does not mean they must. I do not know what conclusion Borel ultimately drew about the reversibility problem. I expect his opinion to be that we have a clear answer, and unsettlingly great room for that answer to be incomplete.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. The next essay should come Friday and will, I hope, be shorter.

Someone Else’s Homework: A Postscript


My friend aced the mathematics final. Not due to my intervention, I’d say; my friend only remembered one question on the exam being much like anything we had discussed recently. Though it was very like one of those, a question about the probability of putting together a committee with none, one, two, or more than two members of particular subgroups. And that one we didn’t even work through; I just confirmed my friend’s guess about what calculation to do. Which is good since that particular calculation is a tedious one that I didn’t want to do. No, my friend aced it by working steadily through the whole term. And yes, asking me for tutoring a couple times, but that’s all right. Small, steady work adds up, in mathematics as with so much else.

Meanwhile may I draw your attention over to my humor blog where last night I posted a bit of silliness about number divisibility. Because I can’t help myself, it does include a “quick” test for whether a number could be divided by 21. It’s in the same spirit as tests for whether a number can be divided by 3 or 9 (add the digits add see whether that sum’s divisible by 3 or 9) or 11 (add or subtract digits, in alternate form, and see whether that sum is divisible by 11). The process I give is correct, which is not to say that anyone would ever use it. Even if they did they’d be better off testing for divisibility by both 3 and 7. And I don’t think I’d use an add-the-digits scheme for 7 either.

Why Shouldn’t We Talk About Mathematics In The Deli Line?


You maybe saw this picture going around your social media a couple days ago. I did, but I’m connected to a lot of mathematics people who were naturally interested. Everyone who did see it was speculating about what the story behind it was. Thanks to the CBC, now we know.

So it’s the most obvious if least excitingly gossip-worthy explanation: this Middletown, Connecticut deli is close to the Wesleyan mathematics department’s office and at least one mathematician was too engrossed talking about the subject to actually place an order. We’ve all been stuck behind people like that. It’s enough to make you wonder whether the Cole slaw there is actually that good. (Don’t know, I haven’t been there, not sure I can dispatch my agent in Groton to check just for this.) The sign’s basically a loving joke, which is a relief. Could be any group of people who won’t stop talking about a thing they enjoy, really. And who have a good joking relationship with the deli-owner.

The CBC’s interview gets into whether mathematicians have a sense of humor. I certainly think we do. I think the habit of forming proofs builds a habit of making a wild assumption and seeing where that gets you, often to a contradiction. And it’s hard not to see that the same skills that will let you go from, say, “suppose every angle can be trisected” to a nonsensical conclusion will also let you suppose something whimsical and get to a silly result.

Dr Anna Haensch, who made the sign kind-of famous-ish, gave as an example of a quick little mathematician’s joke going to the board and declaring “let L be a group”. I should say that’s not a riotously funny mathematician’s joke, not the say (like) talking about things purple and commutative are. It’s just a little passing quip, like if you showed a map of New Jersey and labelled the big city just across the Hudson River as “East Hoboken” or something.

But L would be a slightly disjoint name for a group. Not wrong, just odd, unless the context of the problem gave us good reason for the name. Names of stuff are labels, and so are arbitrary and may be anything. But we use them to carry information. If we know something is a group then we know something about the way it behaves. So if in a dense mass of symbols we see that something is given one of the standard names for groups — G, H, maybe G or H with a subscript or a ‘ or * on top of it — we know that, however lost we might be, we know this thing is a group and we know it should have these properties.

It’s a bit of doing what science fiction fans term “incluing”. That’s giving someone the necessary backstory without drawing attention to the fact we’re doing it. To avoid G or H would be like avoiding “Jane [or John] Doe” as the name for a specific but unidentified person. You can do it, but it seems off.

Finally, What I Learned Doing Theorem Thursdays


Here’s the index to the stuff I posted on them.

The biggest thing I learned from my Theorem Thursdays project was: don’t do this for Thursdays. The appeal is obvious. If things were a little different I’d have no problem with Thursdays. But besides being a slightly-read pop-mathematics blogger I’m also a slightly-read humor blogger. And I try to have a major piece, about seven hundred words that are more than simply commentary on how a comic strip’s gone wrong, ready for Thursday evenings my time.

That’s all my doing. It’s a relic of my thinking that the humor blog should run at least a bit like a professional syndicated columnist’s, with a fixed deadline for bigger pieces. While I should be writing more ahead of deadline than this, what I would do is get to Wednesday realizing I have two major things to write in a day. I’d have an idea for one of them, the mathematics thing, since I would pick a topic the previous Thursday. And once I’ve picked an idea the rest is easy. (Part of the process of picking is realizing whether there’s any way to make seven hundred words about something.) But that’s a lot of work for something that’s supposed to be recreational. Plus Wednesdays are, two weeks a month, a pinball league night.

So Thursday is right out, unless I get better about having first drafts of stuff done Monday night. So Thursday is right out. This has problems for future appearances of the gimmick. The alliterative pull is strong. The only remotely compelling alternative is Theorems on the Threes, maybe one the 3rd, 13th, and 23rd of the month. That leaves the 30th and 31st unaccounted for, and room for a good squabble about whether they count in an “on the threes” scheme.

There’s a lot of good stuff to say about the project otherwise. The biggest is that I had fun with it. The Theorem Thursday pieces sprawled into for-me extreme lengths, two to three thousand words. I had space to be chatty and silly and autobiographic in ways that even the A To Z projects don’t allow. Somehow those essays didn’t get nearly as long, possibly because I was writing three of them a week. I didn’t actually write fewer things in July than I did in, say, May. But it was fewer kinds of things; postings were mostly Theorem Thursdays and Reading the Comics posts. Still, overall readership didn’t drop and people seemed to quite like what I did write. It may be fewer but longer-form essays are the way I should go.

Also I found that people like stranger stuff. There’s an understandable temptation in doing pop-mathematics to look for topics that are automatically more accessible. People are afraid enough of mathematics. They have good reason to be terrified of some topic even mathematics majors don’t encounter until their fourth year. So there’s a drive to simpler topics, or topics that have fewer prerequisites, and that’s why every mathematics blogger has an essay about how the square root of two is irrational and how there’s different sizes to infinitely large sets. And that’s produced some excellent writing about topics like those, which are great topics. They have got the power to inspire awe without requiring any warming up. That’s special.

But it also means they’re hard to write anything new or compelling about if you’re like me, and in somewhere like the second hundred billion of mathematics bloggers. I can’t write anything better than what’s already gone about that. Liouville’s Theorem? That’s something I can be a good writer about. With that, I can have a blog personality. It’s like having a real personality but less work.

As I did with the Leap Day 2016 A To Z project, I threw the topics open to requests. I didn’t get many. Possibly the form gave too much freedom. Picking something to match a letter, as in the A to Z, gives a useful structure for choosing something specific. Pick a theorem from anywhere in mathematics? Something from algebra class? Something mentioned in a news report about a major breakthrough the reporter doesn’t understand but had an interesting picture? Something that you overheard the name of once without any context? How should people know what the scope of it is, before they’ve even seen a sample? And possibly people don’t actually remember the names of theorems unless they stay in mathematics or mathematics-related fields. Those folks hardly need explained theorems with names they remember. This is a hard problem to imagine people having, but it’s something I must consider.

So this is what I take away from the two-month project. There’s a lot of fun digging into the higher-level mathematics stuff. There’s an interest in it, even if it means I write longer and therefore fewer pieces. Take requests, but have a structure for taking them that makes it easy to tell what requests should look like. Definitely don’t commit to doing big things for Thursday, not without a better scheme for getting the humor blog pieces done. Free up some time Wednesday and don’t put up an awful score on Demolition Man like I did last time again. Seriously, I had a better score on The Simpsons Pinball Party than I did on Demolition Man and while you personally might not find this amusing there’s at least two people really into pinball who know how hilarious that is. (The games have wildly different point scorings. This like having a basketball score be lower than a hockey score.) That isn’t so important to mathematics blogging but it’s a good lesson to remember anyway.

Reading the Comics, December 30, 2015: Seeing Out The Year Edition


There’s just enough comic strips with mathematical themes that I feel comfortable doing a last Reading the Comics post for 2015. And as maybe fits that slow week between Christmas and New Year’s, there’s not a lot of deep stuff to write about. But there is a Jumble puzzle.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips gives us someone so wrapped up in measuring data as to not notice the obvious. The obvious, though, isn’t always right. This is why statistics is a deep and useful field. It’s why measurement is a powerful tool. Careful measurement and statistical tools give us ways to not fool ourselves. But it takes a lot of sampling, a lot of study, to give those tools power. It can be easy to get lost in the problems of gathering data. Plus numbers have this hypnotic power over human minds. I understand Lard’s problem.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of December messes with a kid’s head about the way we know 1 + 1 equals 2. The classic Principia Mathematica construction builds it out of pure logic. We come up with an idea that we call “one”, and another that we call “plus one”, and an idea we call “two”. If we don’t do anything weird with “equals”, then it follows that “one plus one equals two” must be true. But does the logic mean anything to the real world? Or might we be setting up a game with no relation to anything observable? The punchy way I learned this question was “one cup of popcorn added to one cup of water doesn’t give you two cups of soggy popcorn”. So why should the logical rules that say “one plus one equals two” tell us anything we might want to know about how many apples one has?

Words: LIHWE, CAQUK, COYKJE, TALAFO. Unscramble to - - - O O, O O - - -, - - - - - O, - - O - O -, and solve the puzzle: 'The math teacher liked teaching addition and subtraction - - - - - - -'.
David L Hoyt and Jeff Knurek’s Jumble for the 28th of December, 2015. The link will probably expire in late January 2016.

David L Hoyt and Jeff Knurek’s Jumble for the 28th of December features a mathematics teacher. That’s enough to include here. (You might have an easier time getting the third and fourth words if you reason what the surprise-answer word must be. You can use that to reverse-engineer what letters have to be in the circles.)

Richard Thompson’s Richard’s Poor Almanac for the 28th of December repeats the Platonic Fir Christmas Tree joke. It’s in color this time. Does the color add to the perfection of the tree, or take away from it? I don’t know how to judge.

A butterfly tells another 'you *should* feel guilty --- the flutter of your wings ended up causing a hurricane that claimed thousands of lives!'
Rina Piccolo filling in for Hilary Price on Rhymes With Orange for the 29th of December, 2015. It’s a small thing but I always like the dog looking up in the title panel for Cartoonist Showcase weeks.

Hilary Price’s Rhymes With Orange for the 29th of December gives its panel over to Rina Piccolo. Price often has guest-cartoonist weeks, which is a generous use of her space. Piccolo already has one and a sixth strips — she’s one of the Six Chix cartoonists, and also draws the charming Tina’s Groove — but what the heck. Anyway, this is a comic strip about the butterfly effect. That’s the strangeness by which a deterministic system can still be unpredictable. This counter-intuitive conclusion dates back to the 1890s, when Henri Poincaré was trying to solve the big planetary mechanics question. That question is: is the solar system stable? Is the Earth going to remain in about its present orbit indefinitely far into the future? Or might the accumulated perturbations from Jupiter and the lesser planets someday pitch it out of the solar system? Or, less likely, into the Sun? And the sad truth is, the best we can say is we can’t tell.

In Brian Anderson’s Dog Eat Doug for the 30th of December, Sophie ponders some deep questions. Most of them are purely philosophical questions and outside my competence. “What are numbers?” is also a philosophical question, but it feels like something a mathematician ought to have a position on. I’m not sure I can offer a good one, though. Numbers seem to be to be these things which we imagine. They have some properties and that obey certain rules when we combine them with other numbers. The most familiar of these numbers and properties correspond with some intuition many animals have about discrete objects. Many times over we’ve expanded the idea of what kinds of things might be numbers without losing the sense of how numbers can interact, somehow. And those expansions have generally been useful. They strangely match things we would like to know about the real world. And we can discover truths about these numbers and these relations that don’t seem to be obviously built into the definitions. It’s almost as if the numbers were real objects with the capacity to surprise and to hold secrets.

Why should that be? The lazy answer is that if we came up with a construct that didn’t tell us anything interesting about the real world, we wouldn’t bother studying it. A truly irrelevant concept would be a couple forgotten papers tucked away in an unread journal. But that is missing the point. It’s like answering “why is there something rather than nothing” with “because if there were nothing we wouldn’t be here to ask the question”. That doesn’t satisfy. Why should it be possible to take some ideas about quantity that ravens, raccoons, and chimpanzees have, then abstract some concepts like “counting” and “addition” and “multiplication” from that, and then modify those concepts, and finally have the modification be anything we can see reflected in the real world? There is a mystery here. I can’t fault Sophie for not having an answer.

Making A Joke Of Entropy


This entered into my awareness a few weeks back. Of course I’ve lost where I got it from. But the headline is of natural interest to me. Kristy Condon’s “Researchers establish the world’s first mathematical theory of humor” describes the results of an interesting paper studying the phenomenon of funny words.

The original paper is by Chris Westbury, Cyrus Shaoul, Gail Moroschan, and Michael Ramscar, titled “Telling the world’s least funny jokes: On the quantification of humor as entropy”. It appeared in The Journal of Memory and Language. The thing studied was whether it’s possible to predict how funny people are likely to find a made-up non-word.

As anyone who tries to be funny knows, some words just are funnier than others. Or at least they sound funnier. (This brings us into the problem of whether something is actually funny or whether we just think it is.) Westbury, Shaoul, Moroschan, and Ramscar try testing whether a common measure of how unpredictable something is — the entropy, a cornerstone of information theory — can tell us how funny a word might be.

We’ve encountered entropy in these parts before. I used it in that series earlier this year about how interesting a basketball tournament was. Entropy, in this context, is low if something is predictable. It gets higher the more unpredictable the thing being studied is. You see this at work in auto-completion: if you have typed in ‘th’, it’s likely your next letter is going to be an ‘e’. This reflects the low entropy of ‘the’ as an english word. It’s rather unlikely the next letter will be ‘j’, because English has few contexts that need ‘thj’ to be written out. So it will suggest words that start ‘the’ (and ‘tha’, and maybe even ‘thi’), while giving ‘thj’ and ‘thq’ and ‘thd’ a pass.

Westbury, Shaoul, Moroschan, and Ramscar found that the entropy of a word, how unlikely that collection of letters is to appear in an English word, matches quite well how funny people unfamiliar with it find it. This fits well with one of the more respectable theories of comedy, Arthur Schopenhauer’s theory that humor comes about from violating expectations. That matches well with unpredictability.

Of course it isn’t just entropy that makes words funny. Anyone trying to be funny learns that soon enough, since a string of perfect nonsense is also boring. But this is one of the things that can be measured and that does have an influence.

(I doubt there is any one explanation for why things are funny. My sense is that there are many different kinds of humor, not all of them perfectly compatible. It would be bizarre if any one thing could explain them all. But explanations for pieces of them are plausible enough.)

Anyway, I recommend looking at the Kristy Condon report. It explains the paper and the research in some more detail. And if you feel up to reading an academic paper, try Westbury, Shaoul, Moroschan, and Ramscar’s report. I thought it readable, even though so much of it is outside my field. And if all else fails there’s a list of two hundred made-up words used in field tests for funniness. Some of them look pretty solid to me.

Reading the Comics, November 18, 2015: All Caught Up Edition


Yes, I feel a bit bad that I didn’t have anything posted yesterday. I’d had a nice every-other-day streak going for a couple weeks there. But I had honestly expected more mathematically themed comic strips, and there just weren’t enough in my box by the end of the 17th. So I didn’t have anything to schedule for a post the 18th. The 18th came through, though, and now I’ve got enough to talk about. And that before I get to reading today’s comics. So, please, enjoy.

Scott Adams’s Dilbert Classics for the 16th of November (originally published the 21st of September, 1992) features Dilbert discovering Bell’s Theorem. Bell’s Theorem is an important piece of our understanding of quantum mechanics. It’s a theorem that excites people who first hear about it. It implies quantum mechanics can’t explain reality unless it can allow information to be transmitted between interacting particles faster than light. And quantum mechanics does explain reality. The thing is, and the thing that casual readers don’t understand, is that there’s no way to use this to send a signal. Imagine that I took two cards, one an ace and one an eight, seal them in envelopes, and gave them to astronauts. The astronauts each travel to ten light-years away from me in opposite directions. (They took extreme offense at something I said and didn’t like one another anyway.) Then one of them opens her envelope, finding that she’s got the eight. Then instantly, even though they’re twenty light-years apart, she knows the other astronaut has an ace in her envelope. But there is no way the astronauts can use this to send information to one another, which is what people want Bell’s Theorem to tell us. (My example is not legitimate quantum mechanics and do not try to use it to pass your thesis defense. It just shows why Bell’s Theorem does not give us a way to send information we care about faster than light.) The next day Dilbert’s Garbageman, the Smartest Man in the World, mentions Dilbert’s added something to Bell’s Theorem. It’s the same thing everybody figuring they can use quantum entanglement to communicate adds to the idea.

Tom Thaves’ Frank and Ernest for the 16th of November riffs on the idea of a lottery as a “tax on people who are bad at math”. Longtime readers here know that I have mixed feelings about that, and not just because I’m wary of cliché. If the jackpot is high enough, you can reach the point where the expectation value of the prize is positive. That is, you would expect to make money if you played the game under the same conditions often enough. But that chance is still vanishingly small. Even playing a million times would not make it likely you would more earn money than you spent. I’m not dogmatic enough to say what your decision should be, at least if the prize is big enough. (And that’s not considering the value placed on the fun of playing. One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket. But many people do enjoy it and I can’t bring myself to say they’re all wrong about feeling enjoyment.)

And it happens that on the 18th Brant Parker and Johnny Hart’s Wizard of Id Classics (originally run the 20th of November, 1965) did a lottery joke. That one is about a lottery one shouldn’t play, except that the King keeps track of who refuses to buy a ticket. I know when we’re in a genre.

Peter Mann’s The Quixote Syndrome for the 16th of November explores something I had never known but that at least the web seems to think is true. Apparently in 1958 Samuel Beckett knew the 12-year-old André Roussimoff. People of my age cohort have any idea who that is when they hear Roussimoff became pro wrestling star André the Giant. And Beckett drove the kid to school. Mann — taking, I think, a break from his usual adaptations of classic literature — speculates on what they might have talked about. His guess: Beckett attempting to ease one of his fears through careful study and mathematical treatment. The problem is goofily funny. But the treatment is the sort of mathematics everyone understands needing and understands using.

John Deering’s Strange Brew for the 17th of November tells a rounding up joke. Scott Hilburn’s The Argyle Sweater told it back in August. I suspect the joke is just in the air. Most jokes were formed between 1922 and 1978 anyway, and we’re just shuffling around the remains of that fruitful era.

Tony Cochrane’s Agnes for the 18th of November tells a resisting-the-word-problem joke. I admit expecting better from Cochrane. But casting arithmetic problems into word problems is fraught with peril. It isn’t enough to avoid obsolete references. (If we accept trains as obsolete. I’m from the United States Northeast, where subways and even commuter trains are viable things.) The problem also has to ask something the problem-solver can imagine wanting to know. It may not matter whether the question asks how far apart two trains, two cars, or two airplanes are, if the student can’t see their distance as anything but trivia. We may need better practice in writing stories if we’re to write story problems.

Reading the Comics, October 29, 2015: Spherical Squirrel Edition


John Zakour and Scott Roberts’s Maria’s Day is going to Sunday-only publication. A shame, but I understand Zakour and Roberts choosing to focus their energies on better-paying venues. That those venues are “writing science fiction novels” says terrifying things about the economic logic of web comics.

This installment, from the 23rd, is a variation on the joke about the lawyer, or accountant, or consultant, or economist, who carefully asks “what do you want the answer to be?” before giving it. Sports are a rich mine of numbers, though. Mostly they’re statistics, and we might wonder: why does anyone care about sports statistics? Once the score of a game is done counted, what else matters? A sociologist and a sports historian are probably needed to give true, credible answers. My suspicion is that it amounts to money, as it ever does. If one wants to gamble on the outcomes of sporting events, one has to have a good understanding of what is likely to happen, and how likely it is to happen. And I suppose if one wants to manage a sporting event, one wants to spend money and time and other resources to best effect. That requires data, and that we see in numbers. And there are so many things that can be counted in any athletic event, aren’t there? All those numbers carry with them a hypnotic pull.

In Darrin Bell’s Candorville for the 24th of October, Lemont mourns how he’s forgotten how to do long division. It’s an easy thing to forget. For one, we have calculators, as Clyde points out. For another, long division ultimately requires we guess at and then try to improve an answer. It can’t be reduced to an operation that will never require back-tracking and trying some part of it again. That back-tracking — say, trying to put 28 into the number seven times, and finding it actually goes at least eight times — feels like a mistake. It feels like the sort of thing a real mathematician would never do.

And that’s completely wrong. Trying an answer, and finding it’s not quite right, and improving on it is perfectly sound mathematics. Arguably it’s the whole field of numerical mathematics. Perhaps students would find long division less haunting if they were assured that it is fine to get a wrong-but-close answer as long as you make it better.

John Graziano’s Ripley’s Believe It or Not for the 25th of October talks about the Rubik’s Cube, and all the ways it can be configured. I grant it sounds like 43,252,003,274,489,856,000 is a bit high a count of possible combinations. But it is about what I hear from proper mathematics texts, the ones that talk about group theory, so let’s let it pass.

The Rubik’s Cube gets talked about in group theory, the study of things that work kind of like arithmetic. In this case, turning one of the faces — well, one of the thirds of a face — clockwise or counterclockwise by 90 degrees, so the whole thing stays a cube, works like adding or subtracting one, modulo 4. That is, we pretend the only numbers are 0, 1, 2, and 3, and the numbers wrap around. 3 plus 1 is 0; 3 plus 2 is 1. 1 minus 2 is 3; 1 minus 3 is 2. There are several separate rotations that can be done, each turning a third of each face of the cube. That each face of the cube starts a different color means it’s easy to see how these different rotations interact and create different color patterns. And rotations look easy to understand. We can at least imagine rotating most anything. In the Rubik’s Cube we can look at a lot of abstract mathematics in a handheld and friendly-looking package. It’s a neat thing.

Scott Hilburn’s The Argyle Sweater for the 26th of October is really a physics joke. But it uses (gibberish) mathematics as the signifier of “a fully thought-out theory” and that’s good enough for me. Also the talk of a “big boing” made me giggle and I hope it does you too.

Izzy Ehnes’s The Best Medicine Cartoon makes, I believe, its debut for Reading the Comics posts with its entry for the 26th. It’s also the anthropomorphic-numerals joke for the week.

Frank Page’s Bob the Squirrel is struggling under his winter fur this week. On the 27th Bob tries to work out the Newtonian forces involved in rolling about in his condition. And this gives me the chance to share a traditional mathematicians joke and a cliche punchline.

The story goes that a dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician, to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”

The punch line has become a traditional joke in the mathematics and science fields. As a joke it comments on the folkloric disconnection between mathematicians and practicality. It also comments on the absurd assumptions that mathematicians and scientists will make for the sake of producing a model, and for getting an answer.

The joke within the joke is that it’s actually fine to make absurd assumptions. We do it all the time. All models are simplifications of the real world, tossing away things that may be important to the people involved but that just complicate the work we mean to do. We may assume cows are spherical because that reflects, in a not too complicated way, that while they might choose to get near one another they will also, given the chance, leave one another some space. We may pretend a fluid has no viscosity, because we are interested in cases where the viscosity does not affect the behavior much. We may pretend people are fully aware of the costs, risks, and benefits of any action they wish to take, at least when they are trying to decide which route to take to work today.

That an assumption is ridiculous does not mean the work built on it is ridiculous. We must defend why we expect those assumptions to make our work practical without introducing too much error. We must test whether the conclusions drawn from the assumption reflect what we wanted to model reasonably well. We can still learn something from a spherical cow. Or a spherical squirrel, if that’s the case.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 28th of October is a binary numbers joke. It’s the other way to tell the joke about there being 10 kinds of people in the world. (I notice that joke made in the comments on Gocomics.com. That was inevitable.)

Eric the Circle for the 29th of October, this one by “Gilly” again, jokes about mathematics being treated as if quite subject to law. The truth of mathematical facts isn’t subject to law, of course. But the use of mathematics is. It’s obvious, for example, in the setting of educational standards. What things a member of society must know to be a functioning part of it are, western civilization has decided, a subject governments may speak about. Thus what mathematics everyone should know is a subject of legislation, or at least legislation in the attenuated form of regulated standards.

But mathematics is subject to parliament (or congress, or the diet, or what have you) in subtler ways. Mathematics is how we measure debt, that great force holding society together. And measurement again has been (at least in western civilization) a matter for governments. We accept the principle that a government may establish a fundamental unit of weight or fundamental unit of distance. So too may it decide what is a unit of currency, and into how many pieces the unit may be divided. And from this it can decide how to calculate with that currency: if the “proper” price of a thing would be, say, five-ninths of the smallest available bit of currency, then what should the buyer give the seller?

Who cares, you might ask, and fairly enough. I can’t get worked up about the risk that I might overpay four-ninths of a penny for something, nor feel bad that I might cheat a merchant out of five-ninths of a penny. But consider: when Arabic numerals first made their way to the west they were viewed with suspicion. Everyone at the market or the moneylenders’ knew how Roman numerals worked, and could follow addition and subtraction with ease. Multiplication was harder, but it could be followed. Division was a diaster and I wouldn’t swear that anyone has ever successfully divided using Roman numerals, but at least everything else was nice and familiar.

But then suddenly there was this influx of new symbols, only one of them something that had ever been a number before. One of them at least looked like the letter O, but it was supposed to represent a missing quantity. And every calculation on this was some strange gibberish where one unfamiliar symbol plus another unfamiliar symbol turned into yet another unfamiliar symbol or maybe even two symbols. Sure, the merchant or the moneylender said it was easier, once you learned the system. But they were also the only ones who understood the system, and the ones who would profit by making “errors” that could not be detected.

Thus we see governments, even in worldly, trade-friendly city-states like Venice, prohibiting the use of Arabic numerals. Roman numerals may be inferior by every measure, but they were familiar. They stood at least until enough generations passed that the average person could feel “1 + 1 = 2” contained no trickery.

If one sees in this parallels to the problem of reforming mathematics education, all I can offer is that people are absurd, and we must love the absurdness of them.

One last note, so I can get this essay above two thousand words somehow. In the 1910s Alfred North Whitehead and Bertrand Russell published the awesome and menacing Principia Mathematica. This was a project to build arithmetic, and all mathematics, on sound logical grounds utterly divorced from the great but fallible resource of human intuition. They did probably as well as human beings possibly could. They used a bewildering array of symbols and such a high level of abstraction that a needy science fiction movie could put up any random page of the text and pass it off as Ancient High Martian.

But they were mathematicians and philosophers, and so could not avoid a few wry jokes, and one of them comes in Volume II, around page 86 (it’ll depend on the edition you use). There, in Proposition 110.643, Whitehead and Russell establish “1 + 1 = 2” and remark, “the above proposition is occasionally useful”. They note at least three uses in their text alone. (Of course this took so long because they were building a lot of machinery before getting to mere work like this.)

Back in my days as a graduate student I thought it would be funny to put up a mock political flyer, demanding people say “NO ON PROP *110.643”. I was wrong. But the joke is strong enough if you don’t go to the trouble of making up the sign. I didn’t make up the sign anyway.

And to murder my own weak joke: arguably “1 + 1 = 2” is established much earlier, around page 380 of the first volume, in proposition *54.43. The thing is, that proposition warns that “it will follow, when mathematical addition has been defined”, which it hasn’t been at that point. But if you want to say it’s Proposition *54.43 instead go ahead; it will not get you any better laugh.

If you’d like to see either proof rendered as non-head-crushingly as possible, the Metamath Proof Explorer shows the reasoning for Proposition *54.43 as well as that for *110.643. And it contains hyperlinks so that you can try to understand the exact chain of reasoning which comes to that point. Good luck. I come from a mathematical heritage that looks at the Principia Mathematica and steps backward, quickly, before it has the chance to notice us and attack.

Reading the Comics, October 5, 2015: Boxes and Hyperboxes Edition


I’ve got more mathematically-themed comic strips than this to write about, but this should do for one day’s postings. Motley did give me the puzzle of figuring out whether the character’s description of a process could be made sensible, which is a bit of extra fun. Boxes and cubes come up in three of the comics, too.

John McPherson’s Close to Home for the 3rd of October drops in the abacus as a backup for the bank’s computers. It’s a cute enough idea. Deep down, I admit, I’m not sure that an abacus would be needed for most of the work a teller has to do during a temporary computer outage, though. Most of the calculations to do would be working out whether there’s enough money in the account to allow a given withdrawal. That’s database-checking, really. Also I’m not sure that’s a model of abacus that’s actually been made, but if I understood what was wanted, then in some ways wasn’t the artwork successful?

Larry Wright’s Motley Classics for the 3rd of October is a rerun from the same day in 1987. Debbie gives the terribly complicated instructions on how to calculate a tip. I’m not sure how tip-calculating got to the pop culture position of “most complicated thing people do with mathematics that isn’t taxes”. Probably that it is a fairly universal need for mathematics that isn’t taxes (and so seasonally bound) explains it. I think she’s describing a valid algorithm, though, if we make some assumptions about her pronouns.

Suppose we start with the price P. Double that and move the decimal one place over, to the left I suppose, and we have 0.20 times P. Suppose that this is the first answer. If we divide this first answer by four, then, this second answer will be 0.05 times P. And subtracting the second answer from the first is, indeed, 0.15 times P, or fifteen percent of the original price. While correct, though, it’s still a lousy algorithm. Too many steps, too much division, and subtraction is a challenge. Taking one-tenth the price plus half a tenth would be numerically identical and less challenging. Taking one-sixth the price would be a division, yes, but get you to near enough fifteen percent with only one move.

Mark Pett’s Lucky Cow for the 4th of October, another rerun, shows off one of the silly semantic-equation games that mathematics majors sometimes play. Forgive them. There’s a similar argument which proves that half a ham sandwich is greater than God. It all amounts to playing on arguments which might (not always!) be correct in form but have things with silly meanings plugged into them.

Stephan Pastis’s Pearls Before Swine for the 4th of October gives Pig the chance to panic. It’s another strip about the difference between what “positive” and “negative” mean in inference testing, and so in medical testing, versus the connotations of “good” and “`bad” they have. I’ve explained this before, in other Reading the Comics essays, so I’ll spare the whole thing. But in short, “positive” in this case means “these test results are so far away from normal values that it strains plausibility to think it’s normal”. “Negative” means “these test results are not so far away from normal values as to strain plausibility to think it’s normal”.

Geoff Grogan’s Jetpack Jr for the 5th of October draws a hypercube as the box little alien Jetpack Junior arrived in. Well, these are some of the common representations of how a four-dimensional cube would look in our three-dimensional space (and that, rendered on a two-dimensional screen). The difficult-to-conceptualize part is that in the cube, seen in the middle third of the strip, every one of the red lines is the same length, and is perpendicular to all its neighbors. The triptych of shapes are all the same four-dimensional cube, too, just rotated along different axes by different amounts.

All my old links to play with hypercube rotations seem to have expired or turn out to be Java applets. Here’s a page that offers a couple of pictures, though. It has a link to an iOS app that should let people play with rotating a four-dimensional hypercube. Might enjoy it. I think this is the first time Jetpack Jr as such has got around here. It used to run as Plastic Babyheads from Outer Space, with a silly overarching story about aliens with plastic baby heads, ah, invading. I don’t think that made the Reading the Comics roster, though, unless some of the aliens mentioned pi, which they might have done.

Charles Brubaker’s Ask A Cat for the 5th of October I think is another debuting strip around here. It’s about the problem of Schrödinger’s Cat, a thought-experiment designed to show we don’t really understand what the conventional mathematical models of quantum mechanics mean. In at least some views, the mathematics of quantum mechanics suggests we could have an apparently ridiculous result: something big, like a cat, that we expect should work like a classical-physics entity, behaving instead like a quantum-mechanical entity, with no definable state. The problem has been with us for eighty years and isn’t well-answered, but that happens. Zeno’s paradoxes have been with us three thousand years and are still showing us things we don’t quite understand about divisibility and continuity.

Anthony Smith’s Learn to Speak Cat for the 5th of October is a completely different cat comic strip that I think is making a debut here. This is more a matter of silly symbolic manipulation than anything serious, though.

Tom Toles’s Randolph Itch, 2 am from the 5th of October is a rerun from 1999. And it shows a soap-bubble cube. Soap bubbles allow for some neat mathematics. They act like animate computers working out the way to enclose a given volume with the least surface area. A web site written by Dr Michael Hutchings at the University of California/Berkeley describes some of the mathematical work involved. Surprising to me is that it was only in the 1970s that the “double bubble conjecture” was proven. That’s a question about how to cover a given volume using two bubbles. The answer is what you might get from playing with soap bubble wands, but it took about a century of working on to prove. Granting, mathematicians did other things with their time, so it wasn’t uninterrupted soap-bubble work. Hutchings includes some review of the field as it existed in the early 2000s, and lists three open problems. The first of them is one that’s understandable even without knowing more mathematical lingo than what R3 is. (And folks who’re hanging around here know that by now.) Also it has pictures of soap bubbles, which are good for a lazy Friday morning.

Reading the Comics, September 6, 2015: September 6, 2015 Edition


Well, we had another of those days where Comic Strip Master Command ordered everybody to do mathematics jokes. I’ll survive.

Henry is frustrated with his arithmetic, until he goes to the pool hall and counts off numbers on those score chips.
Don Trachte’s Henry for the 6th of September, 2015.

Don Trachte’s Henry is a reminder that arithmetic, like so many things, is easier to learn when you’re comfortable with the context. Personally I’ve never understood why some of the discs on pool scoring racks are different colors but imagine it relates to scoring values, somehow. I’ve encountered multiple people who assume I must be good at pool, since it’s all geometry, and what isn’t just geometry is physics. I’ve disappointed them all so far.

Tony Rubino and Gary Markstein’s Daddy’s Home uses arithmetic as an example of joy-crushing school drudgery. It could’ve as easily been asking the capital of Montana.

Scott Adams’s Dilbert Classics, a rerun from the 29th of June, 1992, has Dilbert make a breakthrough in knot theory. The fundamental principle is correct: there are many knots that one could use for tying shoelaces, just as there are many knots that could be used for tying ties. Discovering new ones is a good ways for knot theorists to get a bit of harmless publicity. Nobody needs them. From a knot-theory perspetive it also doesn’t matter if you lace the shoe’s holes crosswise or ladder-style. There are surely other ways to lace the holes, too, but nobody needs them either.

Maria Scrivan’s Half Full uses a blackboard full of mathematical symbols and name-drops Common Core. Fifty years ago this same joke was published, somewhere, with “Now solve it using the New Math” as punchline. Thirty years from now it will run again, with “Now solve it using the (insert name here)” as punchline. Some things are eternal truths.

T Lewis and Michael Fry’s Over The Hedge presents one of those Cretan paradox-style logic problems. Anyway, I choose to read it as such. I’m tickled by it.

And to close things out, both Leigh Rubin’s Rubes and Mikael Wulff and Anders Morgenthaler’s WuMo did riffs on the story of Newton and the falling apple. Is this truly mathematically-themed? Well, it’s tied to the legend of calculus’s origin, so that’s near enough for me.

Reading the Comics, July 24, 2015: All The Popular Topics Are Here Edition


This week all the mathematically-themed comic strips seem to have come from Gocomics.com. Since that gives pretty stable URLs I don’t feel like I can include images of those comics. So I’m afraid it’s a bunch of text this time. I like to think you enjoy reading the text, though.

Mark Anderson’s Andertoons seemed to make its required appearance here with the July 20th strip. And the kid’s right about parentheses being very important in mathematics and “just” extra information in ordinary language. Parentheses as a way of grouping together terms appear as early as the 16th century, according to Florian Cajori. But the symbols wouldn’t become common for a couple of centuries. Cajori speculates that the use of parentheses in normal rhetoric may have slowed mathematicians’ acceptance of them. Vinculums — lines placed over a group of terms — and colons before and after the group seem to have been more popular. Leonhard Euler would use parentheses a good bit, and that settled things. Besides all his other brilliances, Euler was brilliant at setting notation. There are still other common ways of aggregating terms. But most of them are straight brackets or curled braces, which are almost the smallest possible changes from parentheses you can make.

Though his place was secure, Mark Anderson got in another strip the next day. This one’s based on the dangers of extrapolating mindlessly. One trouble with extrapolation is that if we just want to match the data we have then there are literally infinitely many possible extrapolations, each equally valid. But most of them are obvious garbage. If the high temperature the last few days was 78, 79, 80, and 81 degrees Fahrenheit, it may be literally true that we could extrapolate that to a high of 120,618 degrees tomorrow, but we’d be daft to believe it. If we understand the factors likely to affect our data we can judge what extrapolations are plausible and what ones aren’t. As ever, sanity checking, verifying that our results could be correct, is critical.

Bill Amend’s FoxTrot Classics (July 20) continues Jason’s attempts at baking without knowing the unstated assumptions of baking. See above comments about sanity checking. At least he’s ruling out the obviously silly rotation angle. (The strip originally ran the 22nd of July, 2004. You can see it in color, there, if you want to see things like that.) Some commenters have gotten quite worked up about Jason saying “degrees Kelvin” when he need only say “Kelvin”. I can’t join them. Besides the phenomenal harmlessness of saying “degrees Kelvin”, it wouldn’t quite flow for Jason to think “350 degrees” short for “350 Kelvin” instead of “350 degrees Kelvin”.

Nate Frakes’s Break of Day (July 21) is the pure number wordplay strip for this roundup. This might be my favorite of this bunch, mostly because I can imagine the way it would be staged as a bit on The Muppet Show or a similar energetic and silly show. John Atkinson’s Wrong Hands for July 23 is this roundup’s general mathematics wordplay strip. And Mark Parisi’s Off The Mark for July 22nd is the mathematics-literalist strip for this roundup.

Ruben Bolling’s Tom The Dancing Bug (July 23, rerun) is nominally an economics strip. Its premise is that since rational people do what maximizes their reward for the risk involved, then pointing out clearly how the risks and possible losses have changed will change their behavior. Underlying this are assumptions from probability and statistics. The core is the expectation value. That’s an average of what you might gain, or lose, from the different outcomes of something. That average is weighted by the probability of each outcome. A strictly rational person who hadn’t ruled anything in or out would be expected to do the thing with the highest expected gain, or the smallest expected loss. That people do not do things this way vexes folks who have not known many people.

Reading the Comics, July 19, 2015: Rerun Comics Edition


I’m stepping my blog back away from the daily posting schedule. It’s fun, but it’s also exhausting. Sometimes, Comic Strip Master Command helps out. It slowed the rate of mathematically-themed comics just enough.

By this post’s title I don’t mean that my post is a rerun. But several of the comics mentioned happen to be. One of the good — maybe best — things about the appearance of comics on Gocomics.com and ComicsKingdom is that comic strips that have ended, such as Randolph Itch, 2 am or (alas) Cul de Sac can still appear without taking up space. And long-running comic strips such as Luann can have earlier strips be seen to a new audience, again without doing any harm to the newest generation of cartoonists. So, there’s that.

Greg Evans’s Luann Againn (July 13, originally run July 13, 1987) makes a joke of Tiffany not understanding the odds of a contest. That’s amusing enough. Estimating the probability of something happening does require estimating how many things are possible, though, and how likely they are relative to one another. Supposing that every entry in a sweepstakes is equally likely to win seems fair enough. Estimating the number of sweepstakes entries is another problem.

Tom Toles’s Randolph Itch, 2 am (July 13, rerun from July 29, 2002) tells a silly little pirates-and-algebra joke. I like this one for the silliness and the artwork. The only sad thing is there wasn’t a natural way to work equations for a circle into it, so there’d be a use for “r”.

Continue reading “Reading the Comics, July 19, 2015: Rerun Comics Edition”

The Alice in Wonderland Sesquicentennial


I did not realize it was the 150th anniversary of the publication of Alice in Wonderland, which is probably the best-liked piece of writing by any mathematician. At least it’s the only one I can think of that’s clearly inspired a Betty Boop cartoon. I’ve had cause to talk about Carroll’s writing about logic and some other topics in the past. (One was just a day short of three years ago, by chance.)

As mentioned in his tweet John Allen Paulos reviewed a book entirely about Lewis Carroll/Charles Dodgson’s mathematical and logic writing. I was unaware of the book before, but am interested now.

Reading the Comics, June 21, 2015: Blatantly Padded Edition, Part 2


I said yesterday I was padding one mathematics-comics post into two for silly reasons. And I was. But there were enough Sunday comics on point that splitting one entry into two has turned out to be legitimate. Nice how that works out sometimes.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (June 19) uses mathematics as something to heap upon a person until they yield to your argument. It’s a fallacious way to argue, but it does work. Even at a mathematical conference the terror produced by a screen full of symbols can chase follow-up questions away. On the 21st, they present mathematics as a more obviously useful thing. Well, mathematics with a bit of physics.

Nate Frakes’s Break Of Day (June 19) is this week’s anthropomorphic algebra joke.

Life at the quantum level: one subatomic particle suspects the other of being unfaithful because both know he could be in two places at once.
Niklas Eriksson’s Carpe Diem for the 20th of June, 2015.

Niklas Eriksson’s Carpe Diem (June 20) is captioned “Life at the Quantum Level”. And it’s built on the idea that quantum particles could be in multiple places at once. Whether something can be in two places at once depends on coming up with a clear idea about what you mean by “thing” and “places” and for that matter “at once”; when you try to pin the ideas down they prove to be slippery. But the mathematics of quantum mechanics is fascinating. It cries out for treating things we would like to know about, such as positions and momentums and energies of particles, as distributions instead of fixed values. That is, we know how likely it is a particle is in some region of space compared to how likely it is somewhere else. In statistical mechanics we resort to this because we want to study so many particles, or so many interactions, that it’s impractical to keep track of them all. In quantum mechanics we need to resort to this because it appears this is just how the world works.

(It’s even less on point, but Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 21st of June has a bit of riffing on Schrödinger’s Cat.)

Brian and Ron Boychuk’s Chuckle Brothers (June 20) name-drops algebra as the kind of mathematics kids still living with their parents have trouble with. That’s probably required by the desire to make a joking definition of “aftermath”, so that some specific subject has to be named. And it needs parents to still be watching closely over their kids, something that doesn’t quite fit for college-level classes like Intro to Differential Equations. So algebra, geometry, or trigonometry it must be. I am curious whether algebra reads as the funniest of that set of words, or if it just fits better in the space available. ‘Geometry’ is as long a word as ‘algebra’, but it may not have the same connotation of being an impossibly hard class.

Little Iodine does badly in arithmetic in class. But she's very good at counting the calories, and the cost, of what her teacher eats.
Jimmy Hatlo’s Little Iodine for the 18th of April, 1954, and rerun the 18th of June, 2015.

And from the world of vintage comic strips, Jimmy Hatlo’s Little Iodine (June 21, originally run the 18th of April, 1954) reminds us that anybody can do any amount of arithmetic if it’s something they really want to calculate.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney (June 21) is another strip using the idea of mathematics — and particularly word problems — to signify great intelligence. I suppose it’s easier to recognize the form of a word problem than it is to recognize a good paper on the humanities if you only have two dozen words to show it in.

Juba’s Viivi and Wagner (June 21) is a timely reminder that while sudokus may be fun logic puzzles, they are ultimately the puzzle you decide to make of them.

Reading the Comics, June 11, 2015: Bonus Education Edition


The coming US summer vacation suggests Comic Strip Master Command will slow down production of mathematics-themed comic strips. But they haven’t quite yet. And this week I also found a couple comics that, while not about mathematics, amused me enough that I want to include them anyway. So those bonus strips I’ll run at the end of my regular business here.

Bill Hinds’s Tank McNamara (June 6) does a pi pun. The pithon mathematical-snake idea is fun enough and I’d be interested in a character design. I think the strip’s unjustifiably snotty about tattoos. But comic strips have a strange tendency to get snotty about other forms of art.

A friend happened to mention one problem with tattoos that require straight lines or regular shapes is that human skin has a non-flat Gaussian curvature. Yes, that’s how the friend talks. Gaussian curvature is, well, a measure of how curved a surface is. That sounds obvious enough, but there are surprises: a circular cylinder, such as the label of a can, has the same curvature as a flat sheet of paper. You can see that by how easy it is to wrap a sheet of paper around a can. But a ball hasn’t, and you see that by how you can’t neatly wrap a sheet of paper around a ball without crumpling or tearing the paper. Human skin is kind of cylindrical in many places, but not perfectly so, and it changes as the body moves. So any design that looks good on paper requires some artistic imagination to adapt to the skin.

Bill Amend’s FoxTrot (June 7) sets Jason and Marcus working on their summer tans. It’s a good strip for adding to the cover of a trigonometry test as part of the cheat-sheet.

Dana Simpson’s Phoebe and her Unicorn (June 8) makes what I think is its first appearance in my Reading the Comics series. The strip, as a web comic, had been named Heavenly Nostrils. Then it got the vanishingly rare chance to run as a syndicated newspaper comic strip. And newspaper comics page editors don’t find the word “nostril” too inherently funny to pass up. Thus the more marketable name. After that interesting background I’m sad to say Simpson delivers a bog-standard “kids not understanding fractions” joke. I can’t say much about that.

Ruben Bolling’s Super Fun-Pak Comix (June 10, rerun) is an installment of everyone’s favorite literary device model of infinite probabilities. A Million Monkeys At A Million Typewriters subverts the model. A monkey thinking about the text destroys the randomness that it depends upon. This one’s my favorite of the mathematics strips this time around.

And Dan Thompson’s traditional Brevity appearance is the June 11th strip, an Anthropomorphic Numerals joke combining a traditional schoolyard gag with a pun I didn’t notice the first time I read the panel.


And now here’s a couple strips that aren’t mathematical but that I just liked too much to ignore. Also this lets Mark Anderson’s Andertoons get back on my page. The June 10th strip is a funny bit of grammar play.

Percy Crosby’s Skippy (June 6, rerun from sometime in 1928) tickles me for its point about what you get at the top and the bottom of the class. Although tutorials and office hours and extracurricular help, and automated teaching tools, do customize things a bit, teaching is ultimately a performance given to an audience. Some will be perfectly in tune with the performance, and some won’t. Audiences are like that.

What Is 13 Times 7?


AbyssBrain, author of the Mathemagical Site blog on WordPress, commented on that 2-plus-2-equals-5 post a couple days ago with a link to an Abbot and Costello Show sketch, in which Lou Costello proves to the landlord that 13 times 7 equals 28. And better than that, he does it three different ways. I didn’t want something fun as that to languish in the comments, so please, enjoy it here on the front page.

I have always liked comedy sketches about complicated chains of mock reasoning so this sort of thing is designed just for me.

Reading the Comics, February 24, 2014: Getting Caught Up Edition


And now, I think, I’ve got caught up on the mathematics-themed comics that appeared at Comics Kingdom and at Gocomics.com over the past week and a half. I’m sorry to say today’s entries don’t get to be about as rich a set of topics as the previous bunch’s, but on the other hand, there’s a couple Comics Kingdom strips that I feel comfortable using as images, so there’s that. And come to think of it, none of them involve the setup of a teacher asking a student in class a word problem, so that’s different.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (February 21) tells the old joke about how much of fractions someone understands. To me the canonical version of the joke was a Sydney Harris panel in which one teacher complains that five-thirds of the class doesn’t understand a word she says about fractions, but it’s all the same gag. I’m a touch amused that three and five turn up in this version of the joke too. That probably reflects writing necessity — especially for this B.C. the numbers have to be a pair that obviously doesn’t give you one-half — and that, somehow, odd numbers seem to read as funnier than even ones.

Bud Fisher’s Mutt and Jeff (February 21) decimates one of the old work-rate problems, this one about how long it takes a group of people to eat a pot roast. It was surely an old joke even when this comic first appeared (and I can’t tell you when it was; Gocomics.com’s reruns have been a mixed bunch of 1940s and 1950s ones, but they don’t say when the original run date was), but the spread across five panels treats the joke well as it’s able to be presented as a fuller stage-ready sketch. Modern comic strips value an efficiently told, minimalist joke, but pacing and minor punch lines (“some men don’t eat as fast as others”) add their charm to a comic.

Continue reading “Reading the Comics, February 24, 2014: Getting Caught Up Edition”

Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition


So, the mathematics comics ran away from me a little bit, and I didn’t have the chance to write up a proper post on Thursday or Friday. So I’m writing what I probably would have got to on Friday had time allowed, and there’ll be another in this sequence sooner than usual. I hope you’ll understand.

The title for this entry is basically thanks to Zach Weinersmith, because his comics over the past week gave me reasons to talk about Georg Cantor and Bernard Riemann. These were two of the many extremely sharp, extremely perceptive German mathematicians of the 19th Century who put solid, rigorously logical foundations under the work of centuries of mathematics, only to discover that this implied new and very difficult questions about mathematics. Some of them are good material for jokes.

Eric and Bill Teitelbaum’s Bottomliners panel (February 14) builds a joke around everything in some set of medical tests coming back negative, as well as the bank account. “Negative”, the word, has connotations that are … well, negative, which may inspire the question why is it a medical test coming back “negative” corresponds with what is usually good news, nothing being wrong? As best I can make out the terminology derives from statistics. The diagnosis of any condition amounts to measuring some property (or properties), and working out whether it’s plausible that the measurements could reflect the body’s normal processes, or whether they’re such that there just has to be some special cause. A “negative” result amounts to saying that we are not forced to suppose something is causing these measurements; that is, we don’t have a strong reason to think something is wrong. And so in this context a “negative” result is the one we ordinarily hope for.

Continue reading “Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition”

Reading the Comics, October 25, 2014: No Images Again Edition


I had assumed it was a freak event last time that there weren’t any Comics Kingdom strips with mathematical topics to discuss, and which comics I include as pictures here because I don’t know that the links made to them will work for everyone arbitrarily far in the future. Apparently they’re just not in a very mathematical mood this month, though. Such happens; I’m sure they’ll reappear soon enough.

John Zakour and Scott Roberts’ Working Daze (October 22, a “best of” rerun) brings up one of my very many peeves-regarding-pedantry, the notion that you “can’t give more than 100 percent”. It depends on what 100 percent means. The metaphor of “giving 110 percent” is based on the one-would-think-obvious point that there is a standard quantity of effort, which is the 100 percent, and to give 110 percent is to give measurably more than the standard effort. The English language has enough illogical phrases in it; we don’t need to attack ones that are only senseless if you go out to pick a fight with them.

Mark Anderson’s Andertoons (October 23) shows a student attacking a problem with appreciable persistence. As the teacher says, though, there’s no way the student’s attempts at making 2 plus 2 equal 5 is ever not going to be wrong, at least unless we have different ideas about what is meant by 2, plus, equals, and 5. It’s easy to get from this point to some pretty heady territory: since it’s true that two plus two can’t equal five (using the ordinary definitions of these words), then this statement is true not just everywhere in this universe but in all possible universes. This — indeed, all — arithmetic would even be true if there were no universe. But if something can be true regardless of what the universe is like, or even if there is no universe, then how can it tell us anything about the specific universe that actually exists? And yet it seems to do so, quite well.

Tim Lachowski’s Get A Life (October 23) is really an accounting joke, or really more a “taxes they so mean” joke, but I thought it worth mentioning that, really, the majority of the mathematics the world has done have got to have been for the purposes of bookkeeping and accounting. I’m sorry that I’m not better-informed about this so as to better appreciate what is, in some ways, the dark matter of mathematical history.

Keith Tutt and Daniel Saunders’s chipper Lard’s World Peace Tips (October 23) recommends “be a genius” as one of the ways to bring about world peace, and uses mathematics as the comic shorthand for “genius activity”, not to mention sudoku as the comic shorthand for “mathematics”. People have tried to gripe that sudoku isn’t really mathematics; while it’s not arithmetic, though — you could replace the numerals with letters or with arbitrary symbols not to be repeated in one line, column, or subsquare and not change the problem at all — it’s certainly logic.

John Graziano’s Ripley’s Believe It or Not (October 23) besides giving me a spot of dizziness with that attribution line makes the claim that “elephants have been found to be better at some numerical tasks than chimps or even humans”. I can believe that, more or less, though I notice it doesn’t say exactly what tasks elephants are so good (or chimps and humans so bad) at. Counting and addition or subtraction seem most likely, though, because those are processes it seems possible to create tests for. At some stages in human and animal development the animals have a clear edge in speed or accuracy. I don’t remember reading evidence of elephant skills before but I can accept that they surely have some.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 24) applies the tools of infinite series — adding up infinitely many of a sequence of terms, often to a finite total — to parenting and the problem of one kid hitting another. This is held up as Real Analysis — – the field in which you learn why Calculus works — and it is, yeah, although this is the part of Real Analysis you can do in high school.

John Zakour and Scott Roberts’s Maria’s Day (October 25) picks up on the Math Wiz Monster in Maria’s closet mentioned last time I did one of these roundups. And it includes an attack on the “Common Core” standards, understandably: it’s unreasonable to today’s generation of parents that mathematics should be taught any differently from how it was taught to them, when they didn’t understand the mathematics they were being taught. Innovation in teaching never has a chance.

Dave Whamond’s Reality Check (October 25) reminds us that just because stock framing can be used to turn a subtraction problem into a word problem doesn’t mean that it can’t jump all the way out of mathematics into another field.

I haven’t included any comics from today — the 26th of October — in my reading yet but really, what are the odds there’s like a half-dozen comics of obvious relevance with nice, juicy topics to discuss?