Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition
So, the mathematics comics ran away from me a little bit, and I didn’t have the chance to write up a proper post on Thursday or Friday. So I’m writing what I probably would have got to on Friday had time allowed, and there’ll be another in this sequence sooner than usual. I hope you’ll understand.
The title for this entry is basically thanks to Zach Weinersmith, because his comics over the past week gave me reasons to talk about Georg Cantor and Bernard Riemann. These were two of the many extremely sharp, extremely perceptive German mathematicians of the 19th Century who put solid, rigorously logical foundations under the work of centuries of mathematics, only to discover that this implied new and very difficult questions about mathematics. Some of them are good material for jokes.
Eric and Bill Teitelbaum’s Bottomliners panel (February 14) builds a joke around everything in some set of medical tests coming back negative, as well as the bank account. “Negative”, the word, has connotations that are … well, negative, which may inspire the question why is it a medical test coming back “negative” corresponds with what is usually good news, nothing being wrong? As best I can make out the terminology derives from statistics. The diagnosis of any condition amounts to measuring some property (or properties), and working out whether it’s plausible that the measurements could reflect the body’s normal processes, or whether they’re such that there just has to be some special cause. A “negative” result amounts to saying that we are not forced to suppose something is causing these measurements; that is, we don’t have a strong reason to think something is wrong. And so in this context a “negative” result is the one we ordinarily hope for.
Samson’s Dark Side Of The Horse (February 15) shows off a great bit of simplifying the problem. I’m disappointed Samson chose a multiplication problem which wasn’t hard to do in the first place, though; why not 10042 times 80008005 instead?
Zach Weinersmith’s Saturday Morning Breakfast Cereal (February 16) presents “a formative day for Georg Cantor”. It might have gone something like that. In the late 19th and early 20th century came a series of shocking intellectual discoveries about how the world proved to defy common sense, with, for example, relativity smashing our understandings of space and time and simultaneity and such, and quantum mechanics undermining our confidence in determinism and the continuity of paths and the like, and thermodynamics threatening even our sense of how time works.
There were shattering discoveries of the same magnitude happening in mathematics, and Georg Cantor was behind the ones relating to set theory — studying collections of things which share some property (any property), which had been lurking in mathematics for millennia without anyone noticing there were really good, interesting questions to be asked about it — and infinity. He argued, neatly and in the end convincingly, that there must be different kinds of infinity, and these kinds don’t match what might be intuitively obvious distinctions (like, the set of integers versus the set of rational numbers, or the real numbers versus the set of all points in three-dimensional space, or the set of real numbers versus the set of continuous functions). This was hugely controversial at the time, although the controversy has mostly moved out of the world of professional mathematicians and into the world of well-meaning amateurs who find Cantor’s conclusions so strange as to suppose the logic must be wrong.
Saturday Morning Breakfast Cereal pops back in on the 17th, with a reference to the non-trivial zeroes of the Riemann zeta function. This is one of the most famous unsolved problem in mathematics, since Fermat’s Last Theorem and the Four-Color Map theorems have been proved. The Riemann-zeta function itself is this complex-valued function that itself takes in a complex-valued number — that is, you give it a number which may include , that number which, squared, equals -1, and you might get back out of it another complex-valued number. But there are some numbers you can give as input which give back zeroes as outputs. Some of those numbers we know about, and they’re called the trivial solutions because they’re boring. It’s believed that all the interesting solutions, the non-trivial ones, are complex-valued numbers whose real part is equal to one-half, but it’s not proved. And mathematicians have been trying to prove this belief, known as the Riemann Hypothesis, since 1859; it’s a really, deeply hard problem. Of course, mathematicians have to date not put Batman to work on the Riemann Hypothesis.
Bernard Riemann is renowned for many things, including the logical foundation of non-Euclidean geometries, the sorts of geometry that underlies relativity. He also did much to put calculus into its the modern form. When people learn how to integrate functions, they first learn how Riemann showed us how to integrate.
Jason Poland’s Robbie and Bobby (February 18) is a bit of freeform dream-logic which flows from thinking about power. It’s also got one of the better uses for an exponent that I’ve seen in a while.
Bill Amend’s FoxTrot Classics (February 20, rerun) offers the form of a good little probability puzzle: if Peter does figure the chance of a “mad cow burger” (remember when we were tense about mad cow disease?) at one in a zillion, how many burgers could he have before reaching some dangerous threshold? “Dangerous” has to be specified here — a 50 percent chance of eating one? A 10 percent chance? A 50 percent chance of eating two? What is the risk? — although you don’t necessarily need to turn “one in a zillion” to a specific number, and just leave the probability of a mad cow burger to be the abstract number p or, if you’d rather, x.
Bill Whitehead’s Free Range (February 20) uses the page full of mathematical symbols as shorthand for “serious, deep thought about something”, although I can’t help thinking that vanishing after failing to prove one’s existence is the sort of problem for the philosophy department instead. Philosophy and mathematics even use mostly compatible symbols when we get into addressing logic problems. (Though I suppose a philosopher wouldn’t make the mistake of being unable to prove his existence; after all, thinking implies something must be doing the thinking.)
Scott Dikker’s Jim’s Journal (February 20, rerun) shows Dan saying, “Science is like math: it’s black and white — no gray areas”. That’s not really so true, though. If you have a problem set up, you can derive conclusions from that with, usually, pretty rigorous deductions. But the real fun of mathematics is in setting up the problem, which is a matter of judgement. This is especially so in modelling real-world phenomena, since the real world is full of messy confounding factors and you must select what’s probably relevant and what’s probably ignorable. And even if that’s done perfectly the deductions might not be black-and-white: for example, many interesting calculations can only be done by way of making an approximation and showing that the difference between the exact solution and the approximation is tolerably small (ideally, zero).
Of this set I’d say my favorite is the Robbie and Bobby, which as I said just doesn’t make any kind of literal sense but which has this lovely absurdist flow. I also like the way it’s drawn.