I’m curious whether this is going to be the final bunch of mathematics-themed comics for the year 2014. Given the feast-or-famine nature of the strips it’s plausible we might not have anything good through to mid-January, but, who knows? Of the comics in this set I think the first Peanuts the most interesting to me, since it’s funny and gets at something big and important, although the Ollie and Quentin is a better laugh.
Mark Leiknes’s Cow and Boy (December 23, rerun) talks about chaos theory, the notion that incredibly small differences in a state can produce enormous differences in a system’s behavior. Chaos theory became a pop-cultural thing in the 1980s, when Edward Lorentz’s work (of twenty years earlier) broke out into public consciousness. In chaos theory the chaos isn’t that the system is unpredictable — if you have perfect knowledge of the system, and the rules by which it interacts, you could make perfect predictions of its future. What matters is that, in non-chaotic systems, a small error will grow only slightly: if you predict the path of a thrown ball, and you have the ball’s mass slightly wrong, you’ll make a proportionately small error on what the path is like. If you predict the orbit of a satellite around a planet, and have the satellite’s starting speed a little wrong, your prediction is proportionately wrong. But in a chaotic system there are at least some starting points where tiny errors in your understanding of the system produce huge differences between your prediction and the actual outcome. Weather looks like it’s such a system, and that’s why it’s plausible that all of us change the weather just by existing, although of course we don’t know whether we’ve made it better or worse, or for whom.
Charles Schulz’s Peanuts (December 23, rerun from December 26, 1967) features Sally trying to divide 25 by 50 and Charlie Brown insisting she can’t do it. Sally’s practical response: “You can if you push it!” I am a bit curious why Sally, who’s normally around six years old, is doing division in school (and over Christmas break), but then the kids are always being assigned Thomas Hardy’s Tess of the d’Urbervilles for a book report and that is hilariously wrong for kids their age to read, so, let’s give that a pass.
What’s neat about this though is that both Charlie Brown and Sally are right: at least on the introductory level you can’t divide 25 by 50, at least not and get anything more meaningful than “0, with a remainder of 25”, which is an answer so boring we might not bother with it. Many mathematical operations work like this: you can’t subtract a larger whole number from a smaller, can’t divide a larger whole number into a smaller, that sort of thing. The operation being described just doesn’t make sense, and that is phenomenally obvious when you try to do a real-world analogue of the problem, like divide 25 marbles equally to each of 50 people.
And yet if you do try to force it, wonderful things can happen. Subtracting a larger whole number from a smaller forces you to discover, or invent, or at least rationalize the existence of negative numbers, starting from the counting numbers. Trying to divide a bigger integer into a smaller one gives you fractions and rational numbers; both expand your ability to calculate, and count, and estimate, and generally do mathematics. Expanding your idea of what numbers are can be an uneasy thing — I don’t think it’s coincidence that “negative numbers” have a name that with that disapproving-sounding “negative” word in them — but if you can work out consistent rules by which to work you can do things we haven’t before. And we keep doing this: imaginary numbers were added to the mathematical canon when it was noticed that if you took the square root of a negative number, and pretended you don’t notice that there couldn’t be such a thing, that you could use that along the way to get some useful results (originally, from techniques to find the roots of third- and fourth-degree polynomials).
Just because you’d like to do something doesn’t mean you can, certainly; but a lot of great mathematics results from trying to find a way to sensibly do an operation that nobody’s found a way to work out before.
The next day (December 24, or in the original run, December 27, 1967) Sally explains why she isn’t bothered by her bad mathematics scores. I admit I’m inclined more towards Sally than to Charlie Brown here; I would do my best and not worry about my exact grades, although too low a grade was a warning that I wasn’t working well.
Marc Anderson’s Andertoons (December 24) features a student who’s discovered how to make many more possible answers correct. The kid’s got a fair point if all you want to do is make mathematical statements that are correct. The trick, alas, is that we generally want to make mathematical statements that are correct and interesting and relevant. Just throwing a slash through the equals sign makes it easy to make a correct statement, but interesting and relevant are harder challenges.
Missy Meyers’ Holiday Doodles named December 25th as “Grav-Mass”, which I assume is related to it being Sir Isaac Newton’s birthday, probably from some idealistic physics or teaching program trying to make physics seem cooler by hyping it into a holiday. Mathematical history would be noticeably different without Newton, of course: even if his mathematical work were independently invented by others, his presence organized mathematics and physics together in ways that just weren’t imagined before.
Eric the Circle (December 26, this one by ‘AndrewB’) posits Eric trying to improve his diet but finding his circumference-to-diameter ratio stubbornly unchanged. This at least makes me think of such measures of physical fitness as the Body Mass Index, which by itself brings us back to the birth of modern statistics. The Body Mass Index can be traced back to Adolphe Quetelet, one of the first people to bring the data-analysis techniques of astronomy to the study of human beings: their heights, their weights, their fitness — something which became a national obsession in western Europe, especially as military planners noticed that a shockingly high fraction of urban populations were too unhealthy to draft — as well as more sociological matters like relative crime rates and their relation to poverty, education, and alcohol use.
The 19th century got a bit overly optimistic about how to explain very complicated matters with simple numbers, as note how horribly problematic the Intelligence Quotient is. Something like the Body Mass Index is probably useful as a rough guide but it shouldn’t be taken too literally. Unfortunately, these kinds of figures also nearly invariably have decimal points, and so hypnotize people into taking them as carrying more precision than they possibly can.
In Bud Sagendorf’s Popeye (December 27, originally run, I believe, around July 4, 1981) continues Popeye’s adventures in Spinachovia, where the computer Ono has gone and taken over. Among its various dictates it’s abolished school as the computer knows everything so what do the people need to? Bud Sagendorf’s run on Popeye rarely hit on satirical points except by accident, although in this case it kind of does: computers can remember facts and compute arithmetic far faster and better than humans can, so, what do humans need to do that for? For fun, sure; there’s any number of things fun to see even though a machine will do the task better. And yet it’s hard to see how humans can do meaningful thinking without knowing things and being skilled at some computation and such. A car is able to travel much farther and faster and more easily than a human running can, but, there’s still some need for humans to be able to run, isn’t there?
What I’m getting at it is, I used to really impress people with my ability to instantly remember stupid plot points from G.I.Joe cartoons, and now everybody can look them up without even trying, and it’s taken that away from me.
Piers Baker’s Ollie and Quentin (December 28, rerun) plays on a variation of the gambler’s fallacy: if 50 percent of people are happy and 50 percent aren’t, and you know one person is happy, does it follow the other one isn’t? How many people would you have to find who are happy before it becomes ridiculously implausible that the statistic is right, for example?