One is that, as we’ve thought of counting numbers, there is always “one more”. This doesn’t have to be. We could work with perfectly good number systems that have a largest number. We do, in fact. Every computer programming language has some largest integer that it will deal with. If you need a larger number, you have to do something clever. Your clever idea will let you address some range of bigger numbers, but it too will have a maximum. We’ve set those limits large enough that, usually, they’re not an inconvenience. They’re still there.
But those limits are forced on us by the many failings of matter. What when we get just past Plato’s line’s division, into the reasoning of pure mathematics? There we can set up counting numbers. The standard way to do this is to suppose there is a number “1”. And to suppose that, for any counting number we have, there is a successor, a number one-plus-that. If Joey were to ask why there has to be, all Dennis could do is shrug. This makes an axiom out of there always being one more. If you don’t like it, make some other arithmetic. Anyway we only understand any of this using fallible matter, so good luck.
This progression can be heady, though. The counting numbers are probably the most understandable infinitely large set there is. Thinking about them seriously can induce the sort of dizzy awe that pondering Deep Time or the vastness of space can do. That seems a bit above Dennis’s age level, but some people are stricken with the infinite sooner than others are.
Charles Schulz’s Peanuts Begins rerun for the 2nd has Charlie Brown dismiss arithmetic as impractical. It fits the motif of mathematics as an unworldly subject. There’s the common joke that pure mathematics even dreams of being of no use to anyone. Arithmetic, though, has always been a practical subject. It introduces us to many abstract ideas, particularly group theory. This subject looks at what we can do with systems that work like arithmetic without necessarily having numbers, or anything that works with numbers.
And a couple of comic strips mentioned mathematics, although in too slight a way to discuss. Dana Simpson’s Phoebe and her Unicorn on the 30th of April started a sequence in which doodles on Phoebe’s homework came to life. That it’s mathematics homework was mostly incidental. I’m open to the argument that mathematics encourages doodling in a way that, say, spelling does not. I’d also be open to the argument you aren’t doing geometry if you don’t doodle. Anyway. Dan Thompson’s Brevity for the 2nd of May features Sesame Street’s Count von Count. It’s a bit of wordplay on the use of “numbers” for songs. And, of course, the folkloric tradition of vampires as compulsive counters.
And now I’ll cover the handful of comic strips which ran last week and which didn’t fit in my Sunday report. And link to a couple of comics that ultimately weren’t worth discussion in their own right, mostly because they were repeats of ones I’ve already discussed. I have been trimming rerun comics out of my daily reading. But there are ones I like too much to give up, at least not right now.
Bud Blake’s Tiger for the 25th has Tiger quizzing Punkinhead on counting. The younger kid hasn’t reached the point where he can work out numbers without a specific physical representation. It would come, if he were in one of those comics where people age.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is an optimization problem, and an expectation value problem. The wisdom-seeker searches for the most satisfying life. The mathematician-guru offers an answer based in probability and expectation values. List all the possible outcomes, and how probable each are, and how much of the relevant quantity you get (or lose) with each outcome. This is a quite utilitarian view of life-planning. Finding the best possible outcome, given certain constraints, is another big field of mathematics.
John Atkinson’s Wrong Hands for the 26th is a nonsense-equation panel. It’s built on a cute idea. If you do wan to know how many bears you can fit in the kitchen you would need something like this. Not this, though. You can tell by the dimensions. ‘x’, as the area of the kitchen, has units of, well, area. Square feet, or square meters, or square centimeters, or whatever is convenient to measure its area. The average volume of a bear, meanwhile, has units of … volume. Cubic feet, or cubic meters, or cubic centimeters, or the like. The one divided by the other has units of one-over-distance.
And I don’t know what the units of desire to have bears in your kitchen are, but I’m guessing it’s not “bear-feet”, although that would be worth a giggle. The equation would parse more closely if y were the number of bears that can fit in a square foot, or something similar. I say all this just to spoil Atkinson’s fine enough bit of nonsense.
Percy Crosby’s Skippy for the 26th is a joke built on inappropriate extrapolation. 3520 seconds is a touch under an hour. Skippy’s pace, if he could keep it up, would be running a mile every five minutes, 52 seconds. That pace isn’t impossible — I find it listed on charts for marathon runners. But that would be for people who’ve trained to be marathon or other long-distance runners. They probably have different fifty-yard run times.
And now for some of the recent comics that didn’t seem worth their own discussion, and why they didn’t.
Niklas Eriksson’s Carpe Diem for the 20th features reciting the digits of π as a pointless macho stunt. There are people who make a deal of memorizing digits of π. Everyone needs hobbies, and memorizing meaningless stuff is a traditional fanboy’s way of burying oneself in the thing appreciated. Me, I can give you π to … I want to say sixteen digits. I might have gone farther in my youth, but I was heartbroken when I learned one of the digits I had memorized I got wrong, and so after correcting that mess I gave up going farther.
Thaves’s Frank and Ernest for the 18th is a bit of wordplay. There’s something interesting culturally about phrasing “lots of math, but no chemistry”. Algorithms as mathematics makes sense. Much of mathematics is about finding processes to do interesting things. Algorithms, and the mathematics which justifies them, can at least in principle be justified with deductive logic. And we like to think that the universe must make deductive-logical sense. So it is easy to suppose that something mathematical simply must make logical sense.
Chemistry, though. It’s a metaphor for whatever the difference is between a thing’s roster of components and the effect of the whole. The suggestion is that it is mysterious and unpredictable. It’s an attitude strange to actual chemists, who have a rather good understanding of why most things happen. My suspicion is that this sense of chemistry is old, dating to before we had a good understanding of why chemical bonds work. We have that understanding thanks to quantum mechanics, and its mathematical representations.
But we can still allow for things that happen but aren’t obvious. When we write about “emergent properties” we describe things which are inherent in whatever we talk about. But they only appear when the things are a large enough mass, or interact long enough. Some things become significant only when they have enough chance to be seen.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is about mathematicians’ favorite Ancient Greek philosopher they haven’t actually read. (In fairness, Zeno is hard to read, even for those who know the language.) Zeno’s famous for four paradoxes, the most familiar of which is alluded to here. To travel across a space requires travelling across half of it first. But this applies recursively. To travel any distance requires accomplishing infinitely many partial-crossings. How can you do infinitely many things, each of which take more than zero time, in less than an infinitely great time? But we know we do this; so, what aren’t we understanding? A callow young mathematics major would answer: well, pick any tiny interval of time you like. All but a handful of the partial-crossings take less than your tiny interval time. This seems like a sufficient answer and reason to chuckle at philosophers. Fine; an instant has zero time elapse during it. Nothing must move during that instant, then. So when does movement happen, if there is no movement during all the moments of time? Reconciling these two points slows the mathematician down.
Patrick Roberts’s Todd the Dinosaur for the 19th mentions fractions. It’s only used to list a kind of mathematics problem a student might feign unconsciousness rather than do. And takes quite little space in the word balloon to describe. It’d be the same joke if Todd were asked to come up and give a ten-minute presentation on the Battle of Bunker Hill.
Julie Larson’s The Dinette Set for the 19th mentions the Rubik’s Cube. Sometime I should do a proper essay about its mathematics. Any Rubik’s Cube can be solved in at most 20 moves. And it’s apparently known there are some cube configurations that take at least 20 moves, so, that’s nice to have worked out. But there are many approaches to solving a cube, none of which I am competent to do. Some algorithms are, apparently, easier for people to learn, at the cost of taking more steps. And that’s fine. You should understand something before you try to do it efficiently.
Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.
The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.
Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.
And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.
David Hilbert was born in 1862; T-Rex misspoke.
Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.
I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.
John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.
Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.
Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.
There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:
‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So is tiny. And is impossibly tiny. And is ridiculously tiny. You make an error in dropping these squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:
Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.
Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.
I’d like to open today’s installment with a trifle from Thomas K Dye. He’s a friend, and the cartoonist behind the long-running web comic Newshounds, its new spinoff Infinity Refugees, and some other projects.
Q: Have you read the story of Solidus and Virgule? A: Nah, I'm not into slash fiction.
Dye also has a Patreon, most recently featuring a subscribers-only web comic. And he’s good enough to do the occasional bit of spot art to spruce up my work here.
Henry Scarpelli and Craig Boldman’s Archie rerun for the 9th of April, 2018 is, for me, relatable. I think I’ve read off this anecdote before. The first time I took Real Analysis I was completely lost. Getting me slightly less lost was borrowing a library book on Real Analysis from the mathematics library. The book was in French, a language I can only dimly read. But the different presentation and, probably, the time I had to spend parsing each sentence helped me get a basic understanding of the topic. So maybe trying algebra upside-down isn’t a ridiculous idea.
Lincoln Pierce’s Big Nate rerun for the 9th presents an arithmetic sequence, which is always exciting to work with, if you’re into sequences. I had thought Nate was talking about mathematics quizzes but I see that’s not specified. Could be anything. … And yes, there is something cool in finding a pattern. Much of mathematics is driven by noticing, or looking for, patterns in things and then describing the rules by which new patterns can be made. There’s many easy side questions to be built from this. When would quizzes reach a particular value? When would the total number of points gathered reach some threshold? When would the average quiz score reach some number? What kinds of patterns would match the 70-68-66-64 progression but then do something besides reach 62 next? Or 60 after that? There’s some fun to be had. I promise.
Dave Whamond’s Reality Check for the 10th similarly doesn’t depend on what the question is. It happens to be arithmetic, but it could as easily be identifying George Washington or picking out the noun in a sentence.
Leigh Rubin’s Rubes for the 10th riffs on randomness. In this case it’s riffing on the unpredictability and arbitrariness of random things. Random variables are very interesting in certain fields of mathematics. What makes them interesting is that any specific value — the next number you generate — is unpredictable. But aggregate information about the values is predictable, often with great precision. For example, consider normal distributions. (A lot of stuff turns out to be normal.) In that case we can be confident that the values that come up most often are going to be close to the arithmetic mean of a bunch of values. And that there’ll be about as many values greater than the mean as there are less than the mean. And this will be only loosely true if you’ve looked at a handful of values, at ten or twenty or even two hundred of them. But if you looked at, oh, a hundred thousand values, these truths would be dead-on. It’s wonderful and it seems to defy intuition. It just works.
John Atkinson’s Wrong Hands for the 10th is the anthropomorphic numerals joke for the week. It’s easy to think of division as just making numbers smaller: 4 divided by 6 is less than either 4 or 6. 1 divided by 4 is less than either 1 or 4. But this is a bad intuition, drawn from looking at the counting numbers that don’t look boring. But 4 divided by 1 isn’t less than either 1 or 4. Same with 6 divided by 1. And then when we look past counting numbers we realize that’s not always so. 6 divided by ½ gives 12, greater than either of those numbers, and I don’t envy the teachers trying to explain this to an understandably confused student. And whether 6 divided by -1 gives you something smaller than 6 or smaller than -1 is probably good for an argument in an arithmetic class.
Zach Weinersmith, Chris Jones and James Ashby’s Snowflakes for the 11th has an argument about predicting humans mathematically. It’s so very tempting to think people can be. Some aspects of people can. In the founding lore of statistics is the astonishment at how one could predict how many people would die, and from what causes, over a time. No person’s death could be forecast, but their aggregations could be. This unsettles people. It should: it seems to defy reason. It seems to me even people who embrace a deterministic universe suppose that while, yes, a sufficiently knowledgeable creature might forecast their actions accurately, mere humans shouldn’t be sufficiently knowledgeable.
No strips are tagged for the first time this essay. Just noticing.
Comic Strip Master Command hasn’t had many comics exactly on mathematical points the past week. I’ll make do. There are some that are close enough for me, since I like the comics already. And enough of them circle around people being nervous about doing mathematics that I have a title for this edition.
Tony Cochrane’s Agnes for the 24th talks about math anxiety. It’s not a comic strip that will do anything to resolve anyone’s mathematics anxiety. But it’s funny about its business. Agnes usually is; it’s one of the less-appreciated deeply-bizarre comics out there.
Charles Schulz’s Peanuts for the 24th reruns the comic from the 2nd of November, 1970. It has Sally discovering that multiplication is much easier than she imagined. As it is, she’s not in good shape. But if you accept ‘tooty-two’ as another name for ‘four’ and ‘threety-three’ as another name for ‘nine’, why not? And she might do all right in group theory. In that you can select a bunch of things, called ‘elements’, and describe their multiplication to fit anything you like, provided there’s consistency. There could be a four-forty-four if that seems to answer some question.
Hilary Price’s Rhymes with Orange for the 26th is a calculator joke, made explicitly magical. I’m amused but also wonder if those are small wizards or large mushrooms. And it brings up again the question: why do mathematics teachers care about seeing how you got the answer? Who cares, as long as the answer is right? And my answer there is that yeah, sometimes all we care about is the answer. But more often we care about why someone knows the answer is this instead of that. The argument about what makes this answer right — or other answers wrong — should make it possible to tell why. And it often will help inform other problems. Being able to use the work done for one problem to solve others, or better, a whole family of problems, is fantastic. It’s the sort of thing mathematicians naturally try to do.
So last week, for schedule reasons, I skipped the Christmas Eve strips and promised to get to them this week. There weren’t any Christmas Eve mathematically-themed comic strips. Figures. This week, I need to skip New Year’s Eve comic strips for similar schedule reasons. If there are any, I’ll talk about them next week.
John Graziano’s Ripley’s Believe It or Not for the 28th presents the quite believable claim that Professor Dwight Barkley created a formula to estimate how long it takes a child to ask “are we there yet?” I am skeptical the equation given means all that much. But it’s normal mathematician-type behavior to try modelling stuff. That will usually start with thinking of what one wants to represent, and what things about it could be measured, and how one expects these things might affect one another. There’s usually several plausible-sounding models and one has to select the one or ones that seem likely to be interesting. They have to be simple enough to calculate, but still interesting. They need to have consequences that aren’t obvious. And then there’s the challenge of validating the model. Does its description match the thing we’re interested in well enough to be useful? Or at least instructive?
Len Borozinski’s Speechless for the 28th name-drops Albert Einstein and the theory of relativity. Marginal mathematical content, but it’s a slow week.
John Allison’s Bad Machinery for the 29th mentions higher dimensions. More dimensions. In particular it names ‘ana’ and ‘kata’ as “the weird extra dimensions”. Ana and kata are a pair of directions coined by the mathematician Charles Howard Hinton to give us a way of talking about directions in hyperspace. They echo the up/down, left/right, in/out pairs. I don’t know that any mathematicians besides Rudy Rucker actually use these words, though, and that in his science fiction. I may not read enough four-dimensional geometry to know the working lingo. Hinton also coined the “tesseract”, which has escaped from being a mathematician’s specialist term into something normal people might recognize. Mostly because of Madeline L’Engle, I suppose, but that counts.
Samson’s Dark Side of the Horse for the 29th is Dark Side of the Horse‘s entry this essay. It’s a fun bit of play on counting, especially as a way to get to sleep.