On his website, T. W. Körner introduces The Pleasures of Counting as follows:

Longer than “With Rod and Line Through the Gobi Desert”, funnier than “The Wit and Wisdom of the German General Staff” and with more formulae than "A Brief History of Time" [The Pleasures of Counting] was voted Book of the Year by a panel consisting of Mrs E. Körner, Mrs W. Körner, Miss K. Körner and Dr A. Altman (née Körner).

The Pleasures of Counting is hard to categorise. On one hand, the flow and lucidity matches the best works of general non-fiction, even though the book features more tangents than an intro to derivatives course. On the other hand, in contrast to books that merely tell about math, Körner has the gall to make the reader do exercises.

The result is 500 pages of insights, proofs, exercises, and real-world applications about topics ranging from cholera to submarine warfare to weather prediction, all delivered in Körner’s personable style and with a generous heaping of witty anecdotes and the occasional bit of verse. And it is glorious.

Warning: may contain math – but don’t worry

A first question about any book involving math is how much you need to know beforehand for it to be comprehensible.

Most mathematical arguments in The Pleasures of Counting can be followed with straightforward algebra. This does not mean the results themselves are straightforward – some, for instance the derivation of the Lorentz transformation or an outline of Shannon’s theorem, require careful thought and are easy to get lost in. Some exercises also either require or benefit greatly from prior exposure to calculus. However, in general The Pleasures of Counting manages to be both accessible and fairly deep. A casual reader can skip exercises and tricky arguments while still getting the gist, while other readers will find much to dig into in the more intricate proofs and exercises. All notation used is explained in an appendix.

Most people can gain something from this book, and given the breadth of the material, I expect very few will encounter nothing new.

The pleasures of everything under the sun

Körner discusses many common examples of mathematical reasoning and results, including special relativity, Galileo’s arguments about motion, Engima machines, Turing’s work, fractals, sorting algorithms, and the effects of scaling on biology, though always with his own spin on each topic.

I particularly enjoyed Körner’s discussion of dimensional analysis in physics – a fancy way of saying that you figure out what variables some quantity should depend on, fiddle with them until you get an equation where the units (mass, length, time) check out, and then go design bridges with it.

This is an example of the “dangerous but fascinating past-time” of what Körner calls science “in a darkened room” – trying to derive scientific facts from pure thought alone. Science requires both reason and observation; relying on one alone is like trying to walk with one leg. That’s not to say it’s impossible to go places by hopping with one leg: Körner relishes in showing how you can start from small, abstract assumptions and hop over to interesting conclusions, such as why helicopters have long blades or how spacetime works.

The most unique and refreshing parts of The Pleasures of Counting are Körner’s presentation of the works of several somewhat less well-known scientific figures, such as G. I. Taylor, Lewis Fry Richardson, and Patrick Blackett. I have a feeling Körner’s pick of figures to examine is not random – all three are British mathematicians, physicists, or mathematical physicists who lived from the late 1800s to the mid/late-1900s, worked on war-related issues (Blackett was a major advisor on military strategy and operational research in World War II, Taylor participated in the Manhattan Project, and Richardson was an ardent pacifist who was a conscientious objector during World War I and later attempted a mathematical analysis of the causes of war), and studied/taught at Cambridge, like Körner. The timelines make it possible that Taylor and Blackett could have worked in Cambridge at the same time as Körner studied there, though I cannot recall Körner mentioning any personal knowledge of them in _The Pleasures of Counting__.

The pleasures of digression

Körner does not restrict himself to purely mathematical matters. At one point a Socratic dialogue on the axioms of number theory segues into a discussion on the purpose of university:

TEACHER: […] … When Mill wrote On The Subjection of Women [alright, the dialogue may have been going on a slight tangent even before the university stuff], he was consciously following Plato in this, and, still more importantly, in his view that everything is open to question and that positive good may come from rational discussion.

STUART [a student]: And that is what university is all about.

TEACHER: Not really.

STUART: But that is what university ought to be all about.

TEACHER: So you think the taxpayer is parting with large sums of money so that young ladies and gentlemen can sit around discussing life, the universe and everything. You are here to learn mathematics and more mathematics – not to row, play bridge, act or even to find yourselves – and that is what I am going to teach you.

STUART: But, even if that is what the taxpayers want, is it what they ought to get? A university which just trains technicians is not a university; it is a technical college.

TEACHER: Better a good technical college than a corrupt university. What ought you to learn at university besides mathematics?

STUART: Students learn to question received opinions.

TEACHER: So, after I have made you write out 100 times: ‘I must not accept authority,’, what do we do next?

ELEANOR: [another student]: That’s simple. You make us write out: ‘I really, really must not accept authority.”

TEACHER: Besides which, asking questions is the easy bit. It’s finding good answers which is hard. A university is at least as much a repository for the accumulation of human experience and an instrument for passing it on as it is a device for adding to it.

STUART: But just teaching mathematics is not enough. A lot of us will go on to be engineers and managers and will have to take moral decisions. So why don’t you teach us ethics?

TEACHER: But would you actually go to lectures on ethics?

STUART: If the lecturer was good, yes.

TEACHER: But anybody would go to hear Sir Isaiah Berlin lecturing on how to watch paint dry. The question is, would you go listen to your ordinary lecturers talking about ethics?

ELEANOR: Not unless it was for examinations.

STUART: So why not examine it?

TEACHER: What would the examination questions look like? ‘Is it wrong to steal from widows and orphans? Answer yes or no and give brief reasons.’

STUART: There are lots of difficult and interesting moral problems.

TEACHER: Yes, but the problems of the human race are not those of finding the answer to moral problems in hard cases but of acting on the answer in simple ones. American law schools now include courses on ethics, but the only observable result is that the defence in cases of fraud now begins ‘My client’s behaviour has throughout been not merely legal but ethical.’ […] If wisdom were teachable it would surely be our duty to teach it. Since it is not, we simply try to teach mathematics.

Opinionated? Yes. Controversial? Perhaps. Does he have a point? Definitely.

Körner also discusses how to persuade bureaucratic committees (and when to know how to give up), the principles of successful smalltalk, and the philosophical issue of whether and how we should discount future values.

And then, after you’ve been nodding along at one of these digressions for a while, you snap out of a Körner-induced trance, realise you’re halfway through a proof, and that you’ve been enjoying it all the way.

The idle mathematician of an empty day

Ultimately, The Pleasures of Counting is not about the usefulness or applicability of mathematics, but the joy of it. Deriving truths from other truths, or looking at the messiness of the real world and capturing its broad strokes with a few symbols is not just a means to an end but also an art form, a way of thinking, and a purpose in itself.

Körner closes The Pleasures of Counting with the prologue of William Morris’s The Earthly Paradise. This poem is perhaps the best (and certainly most poetic) argument for the importance of “useless” endeavours like math or poetry. My idle blogging cannot beat Morris’s verse, so here is the poem in full:

Of Heaven and Hell I have no power to sing,
I cannot ease the burdens of your fears,
Or make quick-coming death a little thing,
Or bring again the pleasures of past years,
Nor for my words shall ye forget your tears,
Or hope again for aught that I can say,
The idle singer of an empty day.

But rather, when aweary of your mirth,
From full hearts still unsatisfied ye sigh,
And, feeling kindly onto all the earth,
Grudge every minute as it passes by,
Made the more mindful that the sweet days die –
– Remember me a little then I pray,
The idle singer of an empty day.

The heavy trouble, the bewildering care
That weighs us down who live and earn our bread,
These idle verses have no power to bear;
So let me sign of names remembered,
Because they, living not, can ne'er be dead,
Or long time take their memory quite away
From us poor singers of an empty day.

Dreamer of dreams, born out of my due time,
Why should I strive to set the crooked straight?
Let it suffice me that my murmuring rhyme
Beats with light wings against the ivory gate,
Telling a tale not too importunate
To those who in the sleepy region stay,
Lulled by the singer of an empty day.

Folk say, the wizard to a northern king,
At Christmas-tide such wondrous things did show,
That through one window men beheld the spring,
And through another saw the summer glow,
And through a third the fruited vines a-row,
While still unheard, but in its wonted way,
Piped the drear wind of that December day.

So with this Earthly Paradise it is,
If ye read aright and pardon me,
Who strives to build a shadowy isle of bliss
Midmost the beatings of the steely sea,
Where tossed about all hearts of men must be,
Whose ravenous monsters mighty men shall slay,
Not the poor singer of an empty day.