I’m sorry to have fallen silent so long; I was away from home and thought I’d be able to put up a couple of short pieces along the way, and turned out to be rather busy doing other things instead. It’s given me at least one nice problem with dramatic photographs to use in a near-future entry, though, so not all is lost (although I’m trying to think of a way to re-do the work in it that doesn’t involve quite so much algebra; I’m afraid of losing my readers and worse of making a hash of the LaTeX involved). Meanwhile, it’s been surprisingly close to a month since the last summary of comic strips with mathematical themes — I imagine the cartoonists are taking a break on Students In Classroom setups what with it being summer vacation across so much of the United States — so let me return to that.
Morrie Turner’s Wee Pals (July 22) uses the confused-student motif, this one confused by the many ways to make 50 out of adding two numbers. That doesn’t seem like very much, but the Algebra Fact of the Day Twitter feed happened to bring to mind the sequence of Ulam Numbers. These are named for Stanislaw Ulam, famous outside mathematical circles for his work on the Manhattan Project and the creation of the nuclear bomb. He’s also famous in mathematical circles for his part in pioneering the Monte Carlo method, a potent and very flexible technique to find approximate solutions to complicated problems using probabilistic methods.
But among his other work — and he’s got the variety and depth of other work that makes mere mortals like me feel bad — was work in number theory, a small piece of which was drawing attention to a set now called Ulam Numbers. Start the set with the numbers 1 and 2. Then add to this the smallest number that can only be made one way by adding exactly two of the Ulam Numbers; that would be 3. The next would be 4, which you can make from the set 1, 2, 3, using exactly two numbers, only from adding 1 and 3. (Adding 2 and 2 together is an addition using just one of the Ulam Numbers.) 5 doesn’t get in the set, since that can be made by 1 plus 4 or by 2 plus 3; but 6 is the next Ulam Number. There are infinitely many of these numbers (quick, can you think of why?), and you could see them as a kind of a stretch on Fibonacci numbers.
I don’t know that this is a set of any particular use, past being interesting to look at.
Carla Ventresca and Henry Beckett’s On A Claire Day (July 28) puts out what it bills as a “Math Lesson”, to make a joke about losing all the benefits of buying in bulk in trying to actually use the stuff. Naturally, what distracted me in this was whether the numbers as presented actually added up. (The price-per-ounce comes out … not actually near enough equal for my tastes. It’d work better if it were said that only about 20 ounces were spilled, but, “equals same price” in grocery shopping is a term that implies some toleration for not being exactly the same price.)
Jonathan Lemon’s Rabbits Against Magic (August 1) told the Möbius Strip joke. I did chuckle at this one, mostly because of the punch line, which doesn’t exactly flow logically but somehow works, at least for me.
Bud Fisher’s Mutt and Jeff (August 3, rerun) builds several panels out of the setup for a time-to-complete-work word problem. The same joke could probably be dropped into any modern comic strip, although it’d be put as a wiseacre child answering a teacher, and the minor punch line Mutt offers in the second panel would be skipped in order to save on panel space.
Jef Mallett’s Frazz (August 13) builds a word problem from a surplus coffee mug sale that’s part of a summer-long story. (I’m a touch disappointed it’s used for a “Mrs Olsen Is Old” joke, considering Mallett’s been trying to build up the side of Mrs Olsen that isn’t the butt of everyone’s jokes. But the story so far — starting with her rescuing a student from a distracted driver, and here reaching the sale of coffee mugs that students who really liked her, hard as she would be, have given over the years — has given reasons to like her to the audience that isn’t already teachers with wearying students.)
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (August 18) is built on the Paradox of Epimenides the Cretan, the classic problem of self-referential statements, and common savior of humanity from poorly-programmed robots taking over the universe. (It’s also a problem that might be studied in a logic class in the mathematics department, or in a logic class in the philosophy department, so don’t think I’m trying to say people with philosophy blogs that aggregate comic strips shouldn’t also use this strip.)
Much of mathematics is a set of logical deductions, for which we like to suppose statements are true or are false. Self-referential statements offer plenty of ways to make nonsense of that, with Epimenides’s among the pithiest ways to do it. The obvious way to avoid this nonsense is to declare that self-referential statements can’t be treated as being true or false, but that’s not quite satisfying. That is, it’s fine to say the sentence “this sentence is false” is neither true nor false, but it seems unfair to say that means the sentence “this sentence is true” can’t possibly be true, and when we get to considering “this sentence is neither true nor false” we’re sorry to have brought up the subject. At least we realize the subject requires more careful and serious thought.
This demands some particularly serious thought because one of the great realizations of deductive logic was that whether a proposition is true or false ought to depend on the structure of the argument, not on its content. Whether it’s true or false that “the garage is painted red”, in a logical argument, should depend on how the ideas of “garage” and “painted red” relate to one another and not depend on what we see if we go out there and look. Whether the statement is true or not should be the same if we replace the key words with nonsense ones, “the snorble is flongulated”, or (as logicians actually do because they’d sound silly otherwise) with variables, as in “the A is B”. But the structure of the sentence “this sentence is true” is the same as that of “this sentence is false”, and for that matter as that of “the garage is painted red”, so it looks like something in the content of a self-referential statement throws off the game. It’s not obvious what should be right, but then, isn’t the start of reason the realization that what’s obvious is not always right?
And that’s why we can use this to blow up our would-be robot overlords.