As I was saying last week, more things happened than I cared to put into one post.

Five books arrived during the week before last Saturday’s diary post.

Two of them are new books by T.W. Körner: “Naïve Decision Making: mathematics applied to the social world”, and “the Pleasures of Counting”. He is also the author of two other spectacularly good books: “Fourier Analysis” and “a Companion to Analysis: a second first and first second course in analysis”.

I first encountered his “Fourier Analysis”. He begins the introduction, “This book is meant neither as a drill book for the successful nor as a life belt for the unsuccessful student. Rather, it is intended as a shop window for some of the ideas, techniques and elegant results of Fourier analysis.”

It is a wide-ranging book. In particular, he reviews the work of Sir Cyril Burt, knighted for his work in psychology. It has been suggested that not only did he fabricate data, but that he even fabricated co-authors for papers. On the other hand, one of his defenders has said, “I think it is a crime to cast such doubt over a man’s career.”

Along these lines, the authors of “The Bell Curve” declared that they thought all criticism of Burt was politically motivated, and therefore would use his results in their book.

Regardless of which side of the controversy one comes down on, Körner makes it clear that there is mathematical reason to question Burt’s results. No definite answer, but certainly grounds to question.

Having read “The Bell Curve”, I decided that they told a convincing tale – but I wasn’t about to try checking any of their references. Upon seeing convincing evidence that the challenges to Burt were not merely politically motivated, I must dismiss the entire Bell curve book: the authors’ judgment is suspect.

(I had and have no idea how Körner got from fourier analysis to statistics in this book – and I didn’t and don’t care.)

Because the book was so well-written, I kept an eye out for others by him… and the “Companion to Analysis” was every bit as good as his first book. It is, by contrast, a textbook – but a beautifully written one. “This book is intended for those students who might find rigorous analysis a treat.”

Let me illustrate. In his proof of the intermediate value theorem, he says “the method used to prove theorem 1.35 is called ‘ lion hunting’. The method is also called ‘successive bisection’, ‘bisection search’ or simply ‘bisection’.”

I had never heard it called lion hunting, but it is an apt description.

What makes the book great is that he discusses the mathematics. It’s not just a presentation of results.

Interestingly, what I remember from the book is the opening example. Let f be a function from the rationals to the rationals, defined by:

f: Q –> Q

f(x) = -1 if x^2 < 2

f(x) = 1 otherwise.

Then f is continuous and even differentiable; furthermore, its derivative is everywhere 0, but f is not constant. The key, of course, is that f is defined on the rational numbers instead of over the real numbers. And this is how to see where the properties of the real numbers come into calculus as we usually do it.

Anyway, it occurred to me a month ago that I hadn't looked recently to see if he had put out any more books. He is one of the authors on my mathematical "buy without hesitation" list. (John Stillwell is another.)

That's when I found "The Pleasures of Counting" and "Naive Decision Making." The first begins with "Snow on cholera" and "The coming of convoy". The former discusses the analysis by Dr. John Snow of cholera outbreaks in London around 1850. The latter discusses the effectiveness of convoys of surface ships against the depredations of submarine warfare. And that's just the first 38 pages!

Although the main text is relatively devoid of equations, there are exercises in mathematical modelling beginning with convoys.

By contrast, the second book has equations on every page… it looks like a readable introduction to probability and statistics – but I haven't started it yet.

In summary… three of the four books talk about applications – which I love… but even his analysis text, a standard undergraduate course, is entertainingly and refreshingly different. His math is clear and his narrative is engaging.

After I finish the two new ones – usually at night before I fall asleep – I should go back and reread the two old ones.

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