Though the book is called The Pleasures of Counting, it is more about counting's uses. Mathematics are generally only introduced when called for by some real-life problem: how to minimize the dud count of WWI naval shells with minimal waste of the shells; whether to convoy ships or not, and if so, in how large a convoy; why a mouse is merely bruised in a thousand-foot fall while a human is broken and a horse splashes; how to measure the coastline of Britain; that sort of thing. However, the introduction of these interesting problems makes the book quite frustrating, as the problems tend to be accompanied by explanations that think they're being wonderfully lucid but aren't, and I really would like to know the solutions. (And will look for them once I finish this writeup.)
Körner apparently has a reputation as an accessible writer, and at first he certainly seems to be, but this first impression proves illusory. He peppers the book with loads and loads of amusing anecdotes and even jokes, but that's a bit like trying to disguise a less than stellar dinner by offering up a dessert between each course. And why is the meat of the book not up to the standard of the footnotes? I have to think that it's because Körner is doing just what he set out to do: convey information to the reader the way he would to another mathematician, or at least a math class. And that's no good. Because the things mathematicians value — the abstract case over particulars, maximum density in conveying information — are exactly the wrong priorities in communicating with the lay reader, or at least with this lay reader. Here's an example.
In chapter 11, Körner introduces a problem close to my heart, one I've written published articles about: traffic. How do we figure out the true capacity of a road system, accounting for bottlenecks? Where will adding lanes mean freely flowing traffic and where will they just mean the rush hour parking lot will be wider? No motivation problems here: I'd really like to learn the math behind this. Here's how Körner starts. (I'll put his text in boldface.)
We suppose that we have n towns which we label 1, 2, ..., n. Nooooooo! I don't understand the problem yet! You have to give me specifics so I can understand, and then once we're on the same page, then we can talk about the abstract case! No abstractions till I can picture it in my head, and I've never seen a county with n towns in it. Talk about, I dunno, eight towns instead. And for heaven's sake, if we're going to be talking about a flow of 3 trains or 4 trains per hour, do not label the towns 3 and 4 and so on! That's confusing! Give the towns some fricking names. Amsterdam, Berlin, Copenhagen.
The railway line between towns i and j can carry a maximum of cij trains an hour from i to j. And now I understand why Andrew Plotkin is such a fan of monstrously long variable names. Perhaps Körner thinks that this sentence alone will be enough to make me instantly able to recognize what cij means when I see it later on. If so, he is wrong. cij has no self-evident meaning. What's wrong with calling it "capacity" instead of "c"? Especially given that you're still fricking INTRODUCING the problem!
This is no doubt a wonderfully compact way to express this concept mathematically, but it shoves the burden of unpacking it onto me. Sure, when I look at it for a minute, I can dope out what it means: if you add up all the trains that go one way on every piece of track in the rail system, it's the same as what you get when you add up all the trains that go the other way on the same pieces of track. But why is Körner making me do all this mental work? Wouldn't he rather that I put all my brainpower into actually understanding the problem he's setting forth, not the terms in which he's putting it? Evidently not.
One of Körner's many many anecdotes involves Einstein telling a class that a certain mathematician has come up with a much shorter but much less comprehensible method to solve the problem they're about to do, but that since chalk is cheaper than brains, they're going to do it the stupid way. Paper and ink are also cheaper than brains. I would have liked to have seen Körner take Einstein's advice and spell things out more. Talk about the general case after we've got a few concrete cases to refer back to. ("What's cij again? Oh, right, you can get 5 trains per hour from Amherst to Boston.") Break things down step by step, because you can always skip over steps you understand but you can't ask a book to fill in a gap you can't jump. Don't use glyphs when you can use words. But I guess then he wouldn't be talking to mathematicians, who doubtless don't have to unpack a sigma the way I do, and would instead be talking to me, which he doesn't want to do. And yet we also get chapters lamenting mathematicians' isolation and inability to talk to people! Hint: there's more to bridging the gap than opening with a joke.