Reading the Comics, June 11, 2014: Unsound Edition
I can tell the school year is getting near the end: it took a full week to get enough mathematics-themed comic strips to put together a useful bundle of them this time. I don’t know what I’m going to do this summer when there’s maybe two comic strips I can talk about per week and I have to go finding my own initiative to write about things.
Jef Mallet’s Frazz (June 6) is a pun strip, yeah, although it’s one that’s more or less legitimate for a word problem. The reason I have to say “more or less” is that it’s not clear to me whether, per Caulfield’s specification, the amount of ore lost across each Great Lake is three percent of the original cargo or three percent of the remaining cargo. But writing a word problem so that there’s only the one correct solution is a skill that needs development no less than solving word problems is, and probably if we imagine Caulfield grading he’d realize there was an ambiguity when a substantial number of of the papers make the opposite assumption to what he’d had in his mind.
Ruben Bolling’s Tom the Dancing Bug (June 6, and I believe it’s a rerun) steps into some of the philosophically heady waters that one gets into when you look seriously at probability, and that get outright silly when you mix omniscience into the mix. The Supreme Planner has worked out what he concludes to be a plan certain of success, but: does that actually mean one will succeed? Even if we assume that the Supreme Planner is able to successfully know and account for every factor which might affect his success — well, for a less criminal plan, consider: one is certain to toss heads at least once, if one flips a fair coin infinitely many times. And yet it would not actually be impossible to flip a fair coin infinitely many times and have it turn up tails every time. That something can have a probability of 1 (or 100%) of happening and nevertheless not happen — or equivalently, that something can have a probability of 0 (0%) of happening and still happen — is exactly analogous to how a concept can be true almost everywhere, that is, it can be true with exceptions that in some sense don’t matter. Ruben Bolling tosses in the troublesome notion of the multiverse, the idea that everything which might conceivably happen does happen “somewhere”, to make these impossible events all the more imminent. I’m impressed Bolling is able to touch on so much, with a taste of how unsettling the implications are, in a dozen panels and stay funny about it.
Bud Grace’s The Piranha Club (June 9) gives us Enos cheating with perfectly appropriate formulas for a mathematics exam. I’m kind of surprised the Pythagorean Theorem would rate cheat-sheet knowledge, actually, as I thought that had reached the popular culture at least as well as Einstein’s E = mc2 had, although perhaps it’s reached it much as Einstein’s has, as a charming set of sounds without any particular meaning behind them. I admit my tendency in giving exams, too, has been to allow students to bring their own sheet of notes, or even to have open-book exams, on the grounds that I don’t really care whether they’ve memorized formulas and am more interested in whether they can find and apply the relevant formulas. But that doesn’t make me right; I agree there’s value in being able to identify what the important parts of the course are and to remember them well, and even more value in being able to figure out the area of a triangle or a trapezoid from thinking hard about the subject on your own.
Jason Poland’s Robbie and Bobbie (June 10) is looking for philosophy and mathematics majors, so, here’s hoping it’s found a couple more. The joke here is about the classification of logical arguments. A valid argument is one in which the conclusion does indeed follow from the premises according to the rules of deductive logic. A sound argument is a valid argument in which the premises are also true. The reason these aren’t exactly the same thing is that whether a conclusion follows from the premise depends on the structure of the argument; the content is irrelevant. This means we can do a great deal of work, reasoning out things which follow if we suppose that proposition A being true implies B is false, or that we know B and C cannot both be false, or whatnot. But this means we may fill in, Mad-Libs-style, whatever we like to those propositions and come away with some funny-sounding arguments.
So this is how we can have an argument that’s valid yet not sound. It is valid to say that, if baseball is a form of band organ always found in amusement parks, and if amusement parks are always found in the cubby-hole under my bathroom sink, then, baseball is always found in the cubby-hole under my bathroom sink. But as none of the premises going into that argument are true, the argument’s not sound, which is how you can have anything be “valid but not sound”. Identifying arguments that are valid but not sound is good for a couple questions on your logic exam, so, be ready for that.
John Hambrock’s The Brilliant Mind of Edison Lee (June 11) has the brilliant yet annoying Edison trying to prove his genius by calculating precisely where the baseball will drop. This is a legitimate mathematics/physics problem, of course: one could argue that the modern history of mathematical physics comes from the study of falling balls, albeit more of cannonballs than baseballs. If there’s no air resistance and if gravity is uniform, the problem is easy and you get to show off your knowledge of parabolas. If gravity isn’t uniform, you have to show off your knowledge of ellipses. Either way, you can get into some fine differential equations work, and that work gets all the more impressive if you do have to pay attention to the fact that a ball moving through the air loses some of its speed to the air molecules. That said, it’s amazing that people are able to, in effect, work out approximate solutions to “where is this ball going” in their heads, not to mention to act on it and get to the roughly correct spot, lat least when they’ve had some practice.