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.

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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?

Reading The Comics, October 20, 2014: No Images This Edition


Since I started including Comics Kingdom strips in my roundups of mathematically-themed strips I’ve been including images of those, because I’m none too confident that Comics Kingdom’s pages are accessible to normal readers after some time has passed. Gocomics.com has — as far as I’m aware, and as far as anyone has told me — no such problems, so I haven’t bothered doing more than linking to them. So this is the first roundup in a long while I remember that has only Gocomics strips, with nothing from Comics Kingdom. It’s also the first roundup for which I’m fairly sure I’ve done one of these strips before.

Guy Endore-Kaiser and Rodd Perry and Dan Thompson’s Brevity (October 15, but a rerun) is an entry in the anthropomorphic-numbers line of mathematics comics, and I believe it’s one that I’ve already mentioned in the past. This particular strip is a rerun; in modern times the apparently indefatigable Dan Thompson has added this strip to the estimated fourteen he does by himself. In any event it stands out in the anthropomorphic-numbers subgenre for featuring non-integers that aren’t pi.

Ralph Hagen’s The Barn (October 16) ponders how aliens might communicate with Earthlings, and like pretty much everyone who’s considered the question mathematics is supposed to be the way they’d do it. It’s easy to see why mathematics is plausible as a universal language: a mathematical truth should be true anywhere that deductive logic holds, and it’s difficult to conceive of a universe existing in which it could not hold true. I have somewhere around here a mention of a late-19th-century proposal to try contacting Martians by planting trees in Siberia which, in bloom, would show a proof of the Pythagorean theorem.

In modern times we tend to think of contact with aliens being done by radio more likely (or at least some modulated-light signal), which makes a signal like a series of pulses counting out prime numbers sound likely. It’s easy to see why prime numbers should be interesting too: any species that has understood multiplication has almost certainly noticed them, and you can send enough prime numbers in a short time to make clear that there is a deliberate signal being sent. For comparison, perfect numbers — whose factors add up to the original number — are also almost surely noticed by any species that understands multiplication, but the first several of those are 6, 28, 496, and 8,128; by the time 8,128 pulses of anything have been sent the whole point of the message has been lost.

And yet finding prime numbers is still not really quite universal. You or I might see prime numbers as key, but why not triangular numbers, like the sequence 1, 3, 6, 10, 15? Why not square or cube numbers? The only good answer is, well, we have to pick something, so to start communicating let’s hope we find something that everyone will be able to recognize. But there’s an arbitrariness that can’t be fully shed from the process.

John Zakour and Scott Roberts’s Maria’s Day (October 17) reminds us of the value of having a tutor for mathematics problems — if you’re having trouble in class, go to one — and of paying them appropriately.

Steve Melcher’s That Is Priceless (October 17) puts comic captions to classic paintings and so presented Jusepe de Ribera’s 1630 Euclid, Letting Me Copy His Math Homework. I confess I have a broad-based ignorance of art history, but if I’m using search engines correctly the correct title was actually … Euclid. Hm. It seems like Melcher usually has to work harder at these things. Well, I admit it doesn’t quite match my mental picture of Euclid, but that would have mostly involved some guy in a toga wielding a compass. Ribera seems to have had a series of Greek Mathematician pictures from about 1630, including Pythagoras and Archimedes, with similar poses that I’ll take as stylized representations of the great thinkers.

Mark Anderson’s Andertoons (October 18) plays around statistical ideas that include expectation values and the gambler’s fallacy, but it’s a good puzzle: has the doctor done the procedure hundreds of times without a problem because he’s better than average at it, or because he’s been lucky? For an alternate formation, baseball offers a fine question: Ted Williams is the most recent Major League Baseball player to have a season batting average over .400, getting a hit in at least two-fifths of his at-bats over the course of the season. Was he actually good enough to get a hit that often, though, or did he just get lucky? Consider that a .250 hitter — with a 25 percent chance of a hit at any at-bat — could quite plausibly get hits in three out of his four chances in one game, or for that matter even two or three games. Why not a whole season?

Well, because at some point it becomes ridiculous, rather the way we would suspect something was up if a tossed coin came up tails thirty times in a row. Yes, possibly it’s just luck, but there’s good reason to suspect this coin doesn’t have a fifty percent chance of coming up heads, or that the hitter is likely to do better than one hit for every four at-bats, or, to the original comic, that the doctor is just better at getting through the procedure without complications.

Ryan North’s quasi-clip-art Dinosaur Comics (October 20) thrilled the part of me that secretly wanted to study language instead by discussing “light verb constructions”, a grammatical touch I hadn’t paid attention to before. The strip is dubbed “Compressed Thesis Comics”, though, from the notion that the Tyrannosaurus Rex is inspired to study “computationally” what forms of light verb construction are more and what are less acceptable. The impulse is almost perfect thesis project, really: notice a thing and wonder how to quantify it. A good piece of this thesis would probably be just working out how to measure acceptability of a particular verb construction. I imagine the linguistics community has a rough idea how to measure these or else T Rex is taking on way too big a project for a thesis, since that’d be an obvious point for the thesis to crash against.

Well, I still like the punch line.

Reading the Comics, October 14, 2014: Not Talking About Fourier Transforms Edition


I know that it’s disappointing to everyone, given that one of the comic strips in today’s roundup of mathematically-themed such gives me such a good excuse to explain what Fourier Transforms are and why they’re interesting and well worth the time learning. But I’m not going to do that today. There’s enough other things to think about and besides you probably aren’t going to need Fourier Transforms in class for a couple more weeks yet. For today, though, no, I’ll go on to other things instead. Sorry to disappoint.

Glen McCoy and Gary McCoy’s The Flying McCoys (October 9) jokes about how one can go through life without ever using algebra. I imagine other departments get this, too, like, “I made it through my whole life without knowing anything about US History!” or “And did any of that time I spent learning Art do anything for me?” I admit a bias here: I like learning stuff even if it isn’t useful because I find it fun to learn stuff. I don’t insist that you share in finding that fun, but I am going to look at you weird if you feel some sense of triumph about not learning stuff.

Tom Thaves’s Frank and Ernest (October 10) does a gag about theoretical physics, and string theory, which is that field where physics merges almost imperceptibly into mathematics and philosophy. The rough idea of string theory is that it’d be nice to understand why the particles we actually observe exist, as opposed to things that we could imagine existing that that don’t seem to — like, why couldn’t there be something that’s just like an electron, but two times as heavy? Why couldn’t there be something with the mass of a proton but three-quarters the electric charge? — by supposing that what we see are the different natural modes of behavior of some more basic construct, these strings. A natural mode is, well, what something will do if it’s got a bunch of energy and is left to do what it will with it.

Probably the most familiar kind of natural mode is how if you strike a glass or a fork or such it’ll vibrate, if we’re lucky at a tone we can hear, and if we’re really lucky, at one that sounds good. Things can have more than one natural mode. String theory hopes to explain all the different kinds of particles, and the different ways in which they interact, as being different modes of a hopefully small and reasonable variety of “strings”. It’s a controversial theory because it’s been very hard to find experiments that proves, or soundly rules out, a particular model of it as representation of reality, and the models require invoking exotic things like more dimensions of space than we notice. This could reflect string theory being an intriguing but ultimately non-physical model of the world; it could reflect that we just haven’t found the right way to go about proving it yet.

Charles Schulz’s Peanuts (October 10, originally run October 13, 1967) has Sally press Charlie Brown into helping her with her times tables. She does a fair bit if guessing, which isn’t by itself a bad approach. For one, if you don’t know the exact answer, but you can pin down a lower and and upper bound, you’re doing work that might be all you really need and you’re doing work that may give you a hint how to get what you really want. And for that matter, guessing at a solution can be the first step to finding one. One of my favorite areas of mathematics, Monte Carlo methods, finds solutions to complicated problems by starting with a wild guess and making incremental refinements. It’s not guaranteed to work, but when it does, it gets extremely good solutions and with a remarkable ease. Granted this, doesn’t really help the times tables much.

On the 11th (originally run October 14, 1967), Sally incidentally shows the hard part of refining guesses about a solution; there has to be some way of telling whether you’re getting warmer. In your typical problem for a Monte Carlo approach, for example, you have some objective function — say, the distance travelled by something going along a path, or the total energy of a system — and can measure whether an attempted change is improving your solution — say, minimizing your distance or reducing the potential energy — or is making it worse. Typically, you take any refinement that makes the provisional answer better, and reject most, but not all, refinements that make the provisional answer worse.

That said, “Overly-Eight” is one of my favorite made-up numbers. A “Quillion” is also a pretty good one.

Jeff Mallet’s Frazz (October 12) isn’t explicitly about mathematics, but it’s about mathematics. “Why do I have to show my work? I got the right answer?” There are good responses on two levels, the first of which is practical, and which blends into the second: if you give me-the-instructor the wrong answer then I can hopefully work out why you got it wrong. Did you get it wrong because you made a minor but ultimately meaningless slip in your calculations, or did you get it wrong because you misunderstood the problem and did not know what kind of calculation to do? Error comes in many forms; some are boring — wrote the wrong number down at the start and never noticed, missed a carry — some are revealing — doesn’t know the order of operations, doesn’t know how the chain rule applies in differentiation — and some are majestic.

These last are the great ones, the errors that I love seeing, even though they’re the hardest to give a fair grade to. Sometimes a student will go off on a tack that doesn’t look anything like what we did in class, or could have reasonably seen in the textbook, but that shows some strange and possibly mad burst of creative energy. Usually this is rubbish and reflects the student flailing around, but, sometimes the student is on to something, might be trying an approach that, all right, doesn’t work here, but which if it were cleaned of its logical flaws might be a new and different way to work out the problem.

And that blends to the second reason: finding answers is nice enough and if you’re good at that, I’m glad, but is it all that important? We have calculators, after all. What’s interesting, and what is really worth learning in mathematics, is how to find answers: what approaches can efficiently be used on this problem, and how do you select one, and how do you do it to get a correct answer? That’s what’s really worth learning, and what is being looked for when the instruction is to show your work. Caulfield had the right answer, great, but is it because he knew a good way to work out the problem, or is it because he noticed the answer was left on the blackboard from the earlier class when this one started, or is it because he guessed and got lucky, or is it because he thought of a clever new way to solve the problem? If he did have a clever new way to do the problem, shouldn’t other people get to see it? Coming up with clever new ways to find answers is the sort of thing that gets you mathematical immortality as a pioneer of some approach that gets mysteriously named for somebody else.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 14) makes fun of tenure, the process by which people with a long track record of skill, talent, and drive are rewarded with no longer having to fear being laid off or fired except for cause. (Though I should sometime write about Fourier Transforms, as they’re rather neat.)

'Albert, stop daydreaming and eat your soup', which is alphabet soup, apparently, and where you could find E = m c c if you looked just right.
Margaret Shulock’s Six Chix comic for the 14th of October, 2014: Albert Einstein is evoked alongside the origins of his famous equation about c and m and soup and stuff.

Margaret Shulock’s turn at Six Chix (October 14) (the comic strip is shared among six women because … we couldn’t have six different comic strips written and drawn by women all at the same time, I guess?) evokes the classic image of Albert Einstein, the genius, and drawing his famous equation out of the ordinary stuff of daily life. (I snark a little; Shulock is also the writer for Apartment 3-G, to the extent that things can be said to be written in Apartment 3-G.)

Reading the Comics, September 28, 2014: Punning On A Sunday Edition


I honestly don’t intend this blog to become nothing but talk about the comic strips, but then something like this Sunday happens where Comic Strip Master Command decided to send out math joke priority orders and what am I to do? And here I had a wonderful bit about the natural logarithm of 2 that I meant to start writing sometime soon. Anyway, for whatever reason, there’s a lot of punning going on this time around; I don’t pretend to explain that.

Jason Poland’s Robbie and Bobby (September 25) puns off of a “meth lab explosion” in a joke that I’ve seen passed around Twitter and the like but not in a comic strip, possibly because I don’t tend to read web comics until they get absorbed into the Gocomics.com collective.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (September 26) shows how an infinity pool offers the chance to finally, finally, do a one-point perspective drawing just like the art instruction book says.

Bill Watterson’s Calvin and Hobbes (September 27, rerun) wrapped up the latest round of Calvin not learning arithmetic with a gag about needing to know the difference between the numbers of things and the values of things. It also surely helps the confusion that the (United States) dime is a tiny coin, much smaller in size than the penny or nickel that it far out-values. I’m glad I don’t have to teach coin values to kids.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 27) mentions Lagrange points. These are mathematically (and physically) very interesting because they come about from what might be the first interesting physics problem. If you have two objects in the universe, attracting one another gravitationally, then you can describe their behavior perfectly and using just freshman or even high school calculus. For that matter, describing their behavior is practically what Isaac Newton invented his calculus to do.

Add in a third body, though, and you’ve suddenly created a problem that just can’t be done by freshman calculus, or really, done perfectly by anything but really exotic methods. You’re left with approximations, analytic or numerical. (Karl Fritiof Sundman proved in 1912 that one could create an infinite series solution, but it’s not a usable solution. To get a desired accuracy requires so many terms and so much calculation that you’re better off not using it. This almost sounds like the classical joke about mathematicians, coming up with solutions that are perfect but unusable. It is the most extreme case of a possible-but-not-practical solution I’m aware of, if stories I’ve heard about its convergence rate are accurate. I haven’t tried to follow the technique myself.)

But just because you can’t solve every problem of a type doesn’t mean you can’t solve some of them, and the ones you do solve might be useful anyway. Joseph-Louis Lagrange did that, studying the problem of one large body — like a sun, or a planet — and one middle-sized body — a planet, or a moon — and one tiny body — like an asteroid, or a satellite. If the middle-sized body is orbiting the large body in a nice circular orbit, then, there are five special points, dubbed the Lagrange points. A satellite that’s at one of those points (with the right speed) will keep on orbiting at the same rotational speed that the middle body takes around the large body; that is, the system will turn as if the large, middle, and tiny bodies were fixed in place, relative to each other.

Two of these spots, dubbed numbers 4 and 5, are stable: if your tiny body is not quite in the right location that’s all right, because it’ll stay nearby, much in the same way that if you roll a ball into a pit it’ll stay in the pit. But three of these spots, numbers 1, 2, and 3, are unstable: if your tiny body is not quite on those spots, it’ll fall away, in much the same way if you set a ball on the peak of the roof it’ll roll off one way or another.

When Lagrange noticed these points there wasn’t any particular reason to think of them as anything but a neat mathematical construct. But the points do exist, and they can be stable even if the medium body doesn’t have a perfectly circular orbit, or even if there are other planets in the universe, which throws off the nice simple calculations yet. Something like 1700 asteroids are known to exist in the number 4 and 5 Lagrange points for the Sun and Jupiter, and there are a handful known for Saturn and Neptune, and apparently at least five known for Mars. For Earth apparently there’s just the one known to exist, catchily named 2010 TK7, discovered in October 2010, although I’d be surprised if that were the only one. They’re just small.

Professor Peter Peddle has the crazy idea of studying boxing scientifically and preparing strategy accordingly.
Elliot Caplin and John Cullen Murphy’s Big Ben Bolt, from the 23rd of August, 1953 (rerun the 28th of September, 2014).

Elliot Caplin and John Cullen Murphy’s Big Ben Bolt (September 28, originally run August 23, 1953) has been on the Sunday strips now running a tale about a mathematics professor, Peter Peddle, who’s threatening to revolutionize Big Ben Bolt’s boxing world by reducing it to mathematical abstraction; past Sunday strips have even shown the rather stereotypically meek-looking professor overwhelming much larger boxers. The mathematics described here is nonsense, of course, but it’d be asking a bit of the comic strip writers to have a plausible mathematical description of the perfect boxer, after all.

But it’s hard for me anyway to not notice that the professor’s approach is really hard to gainsay. The past generation of baseball, particularly, has been revolutionized by a very mathematical, very rigorous bit of study, looking at questions like how many pitches can a pitcher actually throw before he loses control, and where a batter is likely to hit based on past performance (of this batter and of batters in general), and how likely is this player to have a better or a worse season if he’s signed on for another year, and how likely is it he’ll have a better enough season than some cheaper or more promising player? Baseball is extremely well structured to ask these kinds of questions, with football almost as good for it — else there wouldn’t be fantasy football leagues — and while I am ignorant of modern boxing, I would be surprised if a lot of modern boxing strategy weren’t being studied in Professor Peddle’s spirit.

Eric the Circle (September 28), this one by Griffinetsabine, goes to the Shapes Singles Bar for a geometry pun.

Bill Amend’s FoxTrot (September 28) (and not a rerun; the strip is new runs on Sundays) jumps on the Internet Instructional Video bandwagon that I’m sure exists somewhere, with child prodigy Jason Fox having the idea that he could make mathematics instruction popular enough to earn millions of dollars. His instincts are probably right, too: instructional videos that feature someone who looks cheerful and to be having fun and maybe a little crazy — well, let’s say eccentric — are probably the ones that will be most watched, at least. It’s fun to see people who are enjoying themselves, and the odder they act the better up to a point. I kind of hate to point out, though, that Jason Fox in the comic strip is supposed to be ten years old, implying that (this year, anyway) he was born nine years after Bob Ross died. I know that nothing ever really goes away anymore, but, would this be a pop culture reference that makes sense to Jason?

Tom Thaves’s Frank and Ernest (September 28) sets up the idea of Euclid as a playwright, offering a string of geometry puns.

Jef Mallet’s Frazz (September 28) wonders about why trains show up so often in story problems. I’m not sure that they do, actually — haven’t planes and cars taken their place here, too? — although the reasons aren’t that obscure. Questions about the distance between things changing over time let you test a good bit of arithmetic and algebra while being naturally about stuff it’s reasonable to imagine wanting to know. What more does the homework-assigner want?

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 28) pops back up again with the prospect of blowing one’s mind, and it is legitimately one of those amazing things, that e^{i \pi} = -1 . It is a remarkable relationship between a string of numbers each of which are mind-blowing in their ways — negative 1, and pi, and the base of the natural logarithms e, and dear old i (which, multiplied by itself, is equal to negative 1) — and here they are all bundled together in one, quite true, relationship. I do have to wonder, though, whether anyone who would in a social situation like this understand being told “e raised to the i times pi power equals negative one”, without the framing of “we’re talking now about exponentials raised to imaginary powers”, wouldn’t have already encountered this and had some of the mind-blowing potential worn off.