A couple buildings around town have blackboard paint and a writing prompt on the walls. Here’s one my love and I wandered across the other day while going to Fabiano’s Chocolate for the obvious reason. (The reason was to see their novelty three-foot-tall, 75-pound solid chocolate bunny. Also to buy less huge piles of candy.)
I recognized that mathematics majors had been past. Well, anyone with an interest in popular mathematics might have written they’re grateful for “G. Cantor”. His work’s escaped into the popular imagination, at least a bit. “C. Weirstrauβ”, though, that’s a mathematics major at work.
Karl Weierstrass, the way his name’s rendered in the English-language mathematics books I know, was one of the people who made analysis what it is today. Analysis is, at heart, the study of why calculus works. He attacked the foundations of calculus, which by modern standards weren’t quite rigorous. And he did brilliantly, giving us the modern standards of rigor. He’s terrified generations of mathematics majors by defining what it is for a function to be continuous. Roughly, it means we can draw the graph of a function without having to lift a pencil. He put it in a non-rough manner. He also developed the precise modern idea for what a limit is. Roughly, a limit is exactly what you might think it means; but to be precise takes genius.
Among Weierstrass’s students was Georg Cantor. His is a more familiar name. He proved that just because a set has infinitely many elements in it doesn’t mean that it can’t be quite small compared to other infinitely large sets. His Diagonal Argument shows there must be, in a sense, more real numbers than there are counting numbers. And a child can understand it. Cantor also pioneered the modern idea of set theory. For a while this looked like it might be the best way to understand why arithmetic works like it does. (My understanding is it’s now thought category theory more fundamental. But I don’t know category theory well enough to have an informed opinion.)
The person grateful to Michigan State University basketball I assume wrote that before last Sunday, when the school wrecked so many NCAA tournament brackets.
Terri Libenson’s The Pajama Diaries for the 20th of March mentions, among “reasons for ice cream”, the stress of having “helped with New Math”. It’s a curious reference, to me. I expect it refers to the stress of how they teach arithmetic differently from how it was when you grew up. I expect that feeds any adult’s natural anxiety about having forgot, or never really been good at, arithmetic. Add to that the anxiety of not being able to help your child when you’re called on. And add to that the ever-present fear of looking like a fool. There’s plenty of reason to be anxious.
Still, the reference to “New Math” is curious since, at least in the United States, that refers to a specific era. In the 1960s and 70s mathematics education saw a major revision, called the “New Math”. This revision tried many different approaches, but built around the theory that students should know why mathematics looks like it does. The hope was that in this way students wouldn’t just know what eight times seven was, but would agree that it made sense for this to be 56. The movement is, generally, regarded as a well-meant failure. The reasons are diverse, but many of them amount to it being very hard to explain why mathematics looks like it does. And it’s even harder to explain it to parents, who haven’t gone to school for years and aren’t going to go back to learn eight times seven. And it’s hard for many teachers, who often aren’t specialists in mathematics, to learn eight times seven in a new way either.
Still, the New Math was dead and buried in the United States by the 1980s. And more, Libenson is Canadian. I don’t know what educational fashions, and reform fashions, are like in Canada. I’m curious if Canadian parents or teachers could let me know, what is going on in reforming Canadian mathematics education? Is “New Math” a term of art in Canada now? Or did Libenson pick a term that would communicate efficiently “mathematics but not like I learned it”?
Rudolph Dirk’s The Katzenjammer Kids on the 20th reprinted the strip from the 5th of September, 1943. I mention it here because it contains an example of mathematics talk being used as signifier of great intelligence. The kids expound: “Now, der t’eory uf der twerpsicosis iss dot er sum uf circumvegetatium und der horizontal triggernometry iss equal to der … ” and that’s as far as it needs to go. It isn’t quite mathematics, but it is certainly using a painting of mathematics to make one look bright.
Mark Anderson’s Andertoons got its appearance in here the 20th. It’s got a student resisting the equivalent fractions idea. he kid notes that 1/2 might equal 2/4 and 4/8 and 8/16, but “the ones on the right feel like more bang for your buck”. The kid has a point. These are all the same number. It’s usually easiest to work with the smallest representation that means what you need. But they might convey their meanings differently. I get a different picture, at least, in speaking of “half the class not being done with the assignment” versus “16 of the 32 students aren’t done with the assignment”.
Charlie Podrebarac’s CowTown for the 20th of March claims Charlie could “literally paper the Earth” with losing NCAA brackets. As I make it out, he’s right. There are 263 possible NCAA brackets, because there are 63 matches in the college basketball tournament. All but one of these are losing. If each bracket fits on one sheet of paper — well, how big is a sheet of paper? If each bracket is on a sheet of A4-size paper, then, each page is 1/16th of a square meter. This is easy to work with. Unfortunately, if Charlie cares about the NCAA college basketball tournament, he’s probably in the United States. So he would print out on paper that’s 8 ½ inches by 11 inches. That’s not quite 1/16th of a square meter or any other convenient-to-work-with size. It’s 93.5 square inches but what good is that?
Well, I will pretend that the 8 ½ by 11 inch paper is close enough to A4. It’s going to turn out not to matter. 263 is 9,223,372,036,854,775,808. Subtract one and we have 9,223,372,036,854,775,807. Big difference. Multiply this by one-sixteenth of a square meter and we have about 576,460,752,000,000,000 square meters of paper. I’m rounding off because it is beyond ridiculous that I didn’t before. The surface area of the Earth is about 510,000,000,000,000 square meters. So if Bob picked every possible losing bracket he could indeed literally paper the Earth a thousand times over and have some paper to spare.
Ruben Bolling’s Super-Fun-Pak Comix for the 21st of March is a Guy Walks Into A Bar that depends on non-base-ten arithmetic for its punch line. I’m amused. I learned about different bases as a kid, in the warm glow of the New Math. The different bases and how they changed what arithmetic looked like enchanted me. Today I know there’s not much need for bases besides ten (normal mathematics), two (used by computers), and sixteen (used by people dealing with computers). (Base sixteen converts easily to base two, so people can understand what the computer is actually doing, while being much more compact, so people don’t have to write out prodigiously long sequences of digits.) But for a while there you can play around with base five or base twelve or, as a horse might, base four. It can help you better appreciate how much thought there is behind something as straightforward as “10”.
The United States is about to spend a good bit of time worrying about the NCAA men’s basketball tournament. It’s a good distraction from the women’s basketball tournament and from the National Invitational Tournament. Last year I used this to write a couple essays that stepped into information theory. Nobody knowledgeable in information theory has sent me threatening letters since. So since the inspiration is back in season I’d like to bring them to your attention again:
But How Interesting Is A Real Basketball Tournament? Because I started out assuming that games were perfectly even match ups either team was likely to win. This isn’t so. If we grant that a number-16 seed is almost sure to lose to a number-1 seed, how does the information content change?
Yes, I can hear people snarking, “not even the tiniest bit”. These are people who think calling all athletic contests “sportsball” is still a fresh and witty insult. No matter; what I mean to talk about applies to anything where there are multiple possible outcomes. If you would rather talk about how interesting the results of some elections are, or whether the stock market rises or falls, whether your preferred web browser gains or loses market share, whatever, read it as that instead. The work is all the same.
To talk about quantifying how interesting the outcome of a game (election, trading day, whatever) means we have to think about what “interesting” qualitatively means. A sure thing, a result that’s bound to happen, is not at all interesting, since we know going in that it’s the result. A result that’s nearly sure but not guaranteed is at least a bit interesting, since after all, it might not happen. An extremely unlikely result would be extremely interesting, if it could happen.