Book Review: The Book of Why

The Book of Why: The New Science of Cause and Effect
Judea Pearl
Mathematics, computers, statistics

The Book of Why reminded me strongly of The Evolution of Beauty.

• The author has a Theory, referred to with Capital Letters (here, the Causal Revolution; there, that Beauty Happens).
• The Theory was actually discovered long ago, but it has been forgotten.
• Mainstream investigators have sneered at, pooh-poohed, and generally neglected the Theory.
• Fie on them.
• They are enslaved to their own outdated notions, which lead them to more and more outlandish convolutions to explain things that the Theory explains quite simply.
• The Theory is nothing less than revolutionary, and has profound implications.
• The assertions of significance are maybe a little more extravagant than the book can establish.

There's even a shared villain of sorts, the statistician R. A. Fisher.

Among the differences is that The Book of Why is more technical. I won't say it's written for a specialist audience, but to get a lot out of it you should have at least a basic understanding of probability and statistics. (Bonus points for knowing what Bayes' Theorem is.) You'll need to do some simple probability math if you want to verify what Pearl is saying. 

Put it another way: if you're not familiar with the stock phrase "correlation is not causation," this isn't the book for you--because this is exactly what Pearl is arguing about. Specifically, The Book of Why argues for the power of making inferences based on causal diagrams, and demonstrates rigorous ways to manipulate them to draw powerful conclusions. 

This sounds hazy, so let's go with an example from the book: the low-birth-weight paradox. We've all learned that smoking is bad, and that it's especially bad for pregnant mothers. And yet: babies with low birth weight do better if their mothers were smokers. This isn't a fluke; the statistics establish correlation quite firmly. What gives?

What gives, says Pearl, is that we're settling for a correlational answer when we need a causal one. Specifically, we need to understand that low birth weight may have different causes. Smoking can cause it. But so can developmental defects, or malnutrition. What the paradox show is that babies whose birth weight is low because their mothers smoked do better than babies whose birth weight is low because of other, much more serious conditions. Which makes perfect sense.

Interesting stuff. On the other hand, classical statisticians have good ways to talk about this sort of effect without resorting to Pearl's "Causal Revolution." Maybe it's clearer and simpler with causal calculus--I'm inclined to believe that--but The Book of Why rather implies a stronger claim.

Moreover, Pearl seems to tack sideways around one of the standard arguments against causal thinking. When you create a causal model, you're making assumptions. All the graph-theoretical rigor in the world won't help you answer questions if your graph is wrong--if, say, you assume that plowing the prairie causes rain (don't laugh, people did this), you'll have an incorrect diagram. Sure, this is often trivial--snowy weather causes traffic accidents, not the other way around--but when it's that trivial, why do you need a causal model in the first place?

The power of standard statistics is to tease out correlations that you didn't expect. That's why statisticians, AI researchers, and machine-learning people love it: you dump in a bunch of data, push a button, and poof! you learn something. (In practice, I haven't found it to work that way, but the notion is seductive.) Causal modeling seems like an interesting and powerful approach to quantifying what it is that you've learned. As to whether it merits the designation of "revolution," though, I'm still agnostic.

Book Review: When Einstein Walked With Gödel

When Einstein Walked With Gödel: Excursions to the Edge of Thought
Jim Holt
Science, mathematics, philosophy

I really like Jim Holt's 2012 book Why Does the World Exist? It's both a meaty intellectual challenge and a playful, engaging read. When Einstein Walked With Gödel is equally engaging, but less meaty. The difference is simply that Why Does the World Exist is a focused collection that deeply explores a single topic, while When Einstein Walked With Gödel is a diffuse collection that shallowly explores many topics.

This isn't a diss. Jim Holt is a very good writer, and his essays are not unreminiscent of Carl Sagan in both their discursiveness and their humanity. He's especially good at weaving together biography and abstract ideas. These collected essays cover twenty years of writing, though; inevitably there is some overlap. The individual pieces are excellent, but too short to give more than an overview of their subjects. The thing as a whole is brilliant, but fragmented--kaleidoscopic isn't too strong a term.

And it's nobody's fault but my own that my brain insists on setting the book's title to the tune of "When Irish Eyes Are Smiling."

Book Review: Beyond Infinity

Beyond Infinity: An Expedition to the Outer Limits of Mathematics
Eugenia Cheng
Mathematics

This reads like My Big Little Book of Advanced Mathematics. It would be a good introduction for someone who's seriously intimidated by math; it's engagingly written, sprinkled with personal anecdotes and useful analogies. If you're mathematically literate, however, this is at best a quick diversion.

---

Everything And More covers very much the same territory. It's by David Foster Wallace, so it decidedly does not read like a book for middle schoolers.

Book Review: Descartes' Secret Notebook

Descartes' Secret Notebook: A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe
Amir Aczel
Biography, mathematics, philosophy

I've enjoyed many of Amir Aczel's books, but this one was a letdown. Mostly it's a capsule biography of René Descartes, written in young-adult prose. The titular notebook is only discussed for maybe twenty-five pages out of 200+, and its contents don't prove to be tremendously revelatory. The explanations of Descartes' union of algebra and geometry were good; there should have been more of them, though. Aczel also wastes a lot of time on what seem to me to be decidedly peripheral questions--whether Descartes was a Rosicrucian, for example. I expect the book would be better for younger readers, or readers with virtually no familiarity with the subject matter.

---

The same author's Pendulum: Léon Foucault and the Triumph of Science is one of his many good books. For a good Descartes book, try Russell Shorto's Descartes' Bones.

Book Review: Blackett's War

Blackett's War: The Men Who Defeated the Nazi U-Boats
Stephen Budiansky
History, biography, mathematics

Round about page 100 of Blackett's War I started thinking: "This is great stuff, but it's all background. Where's the foreground?" It turns out that the whole book is like that. The facts are fascinating, the writing is excellent, the detail is good, and the math is accessible; it's just that the book reads as a collection of anecdotes rather than as a whole.

This is a case where a simple chronological organization doesn't necessarily work well. The eponymous Patrick Blackett sidles into and out of the story, never remaining on-stage for long. Other characters do similarly, as do a succession of political intrigues, military-technical debates, and side stories (including, naturally, the many-times-told Enigma tale). If I'd been editing this material, I might have advised a thematic approach. What Blackett and his colleagues accomplished is really interesting in its own right: the ideas, not the people, are arguably the main characters.

What makes those ideas so interesting is that, in many cases, the key insight was that someone had to ask the question. Once the question was asked, answering it didn't require a Bletchley Park--just basic math and statistics--and yet the answers were no less consequential than the work of the codebreakers. Do larger convoys require substantially more protection? (No.) Why is the line to clean mess kits so long? (Washing takes longer than rinsing; you need three wash tubs and one rinse tub, not two of each.) What's the right depth setting for an air-dropped depth charge? (Shallow. Once your target sub has reached 100' depth, it's also had time to turn, so it won't be where you're aiming anyway.) Should we use bombers to attack cities, or to attack submarines? (Submarines.)

That last one was an obvious fact, by the way, which was ignored. That's the other reason the idea content of Blackett's War needs to be promoted: the staggering arrogance, incompetence, and all-around stupidity of the military men whose job it was to win the war, but whose hobby was insisting that everything they already knew was correct and that being smart was bad. Sir Henry Tizard, for example, had a meeting with a senior naval officer who sniffily explained that it simply wasn't possible to put radar on warships, my dear fellow . . . because there was no space for another aerial on the mast.

The upshot, then, is that I was fully interested and engaged--but I'm not sure that a general reader would be. I'd recommend Blackett's War for readers who have an interest in the Battle of the Atlantic and/or mathematics and/or military stupidity. If you're expecting a character-driven biography, you may be disappointed.

Book Review: The Man Who Knew Infinity

The Man Who Knew Infinity: A Life of the Genius Ramanujan
Robert Kanigel
Biography, Mathematics

There's always a debate about genius. Are there people who just have an inexplicable gift for something? Or is it simply a matter of getting ten thousand hours of practice?

To some extent this is a false dichotomy. To put in 10,000 hours on anything, particularly if you do it as a youth, you've got to be abnormally attracted to that thing. All the same, my personal intuition leans toward the inexplicable-gift side. I thought about that a lot while reading The Man Who Knew Infinity.

I'm not bad at mathematics. I use basic probability and statistics all the time. I have a degree in a science (astronomy) that uses math pretty heavily. I sometimes even do math-oriented puzzles for fun. But numbers don't "speak to me". I don't love mathematics.

Ramanujan--born poor in India in 1887, with little or no formal training, ignored for years by the mathematical establishment--did love mathematics. "Love" may not be a strong enough word. He was a smart young man, who did well in all his classes ... until he got hold of a mathematics book. After that, he wouldn't--maybe couldn't--study anything else. His gift eventually made him famous; by bringing him to England, it may also have killed him.

It's easy to mythologize someone with such an outsized talent and moving life story. The Man Who Knew Infinity does a terrific job of humanizing Ramanujan. It's less successful at conveying the details of his mathematics, but I'm not sure how you could do better. Kanigel chose to assume that his readers don't have even a high-school level of numeracy, which irritated me, but to make any other assumption would probably have irritated a lot of other people. It would have been nice to get the actual details even one of Ramanujan's extraordinary proofs--as things stand, we have to take it on faith that they really were as strange, as beautiful, and as unexpected as other mathematicians say they are--but I don't know that that would be possible.

So the book is, perhaps, as successful as it could be. By focusing on Ramanujan as a person--his religious devotion, his friendship with the English mathematician G. H. Hardy, his sternly vegetarian principles, his homesickness--it makes him accessible. Or maybe it only makes him seem accessible. Ramanujan's torrid love affair with numbers is not something that can be conveyed in text, nor experienced by most mortals.

---

The Man Who Knew Infinity has a fair amount in common with Andrew Hodges's insightful Alan Turing: The Enigma. The latter was the loose basis for the excellent (though quite inaccurate) film The Imitation Game; and, as it happens, a film based on The Man Who Knew Infinity will be released soon.

Book Review: Everything And More

Everything and More: A Compact History of Infinity
David Foster Wallace
Mathematics

DISCLAIMER: David Foster Wallace graduated from Amherst College a year before me. As far as I know, our paths never crossed.

It's hard to assess Everything and More. It's probably too consciously literary for most mathematicians, and it's certainly too mathematical for most humanists. David Foster Wallace set out to write a book about advanced mathematics that would be reasonably accurate, stylishly written, and demanding of no more than high-school math. It's no surprise that he didn't fully succeed; the wonder is that he succeeded at all.

The book starts zippily enough. If you're not actually math-phobic, you can probably get through the first 100-150 pages pretty easily, particularly if you're willing to skim some bits (which I was not, but never mind). By the time we get to EMERGENCY GLOSSARY II, that's not really an option. If you want to profit from this book, you won't need to do differential equations, but you will need slow down and seriously think through some stuff.

If you're willing to do that, your reward will be a not-totally-rigorous-but-mathematically-informed understanding of some deep concepts, such as:

• Why some infinities are larger than other infinities
• Zeno's Paradoxes
• What set theory really is, and what it's good for, and why it matters
• Some mathematical proofs that are both brilliant and quite simple

So that's half the challenge. The other half is that Wallace is absolutely writing to impress. Which is not necessarily a bad thing. I much prefer his literary pyrotechnics to the turgid Academic-ese that often clogs the arteries of intellectually-challenging books. On the other hand, Wallace's consciously literary register--I'm tempted to say "postmodern", except that I'm never sure that "postmodern" means anything--is equally disconcerting if you're expecting the straight, pellucid prose of (say) John Allen Paulos, or even the New Yorkerly eloquence of a John McPhee. Everything and More brims with interpolations, with irony, and with a certain self-aware hipness. You Have Been Warned.

To bracket the target audience, then, I'll give two comparisons. 

• Subject-matter-wise, there's a pretty good overlap with Douglas Hofstadter's classic Gödel, Escher, Bach. Some of the same actors, and some of the same math concepts, pop up in each.
• Stylistically, it pairs well with Neal Stephenson, most particularly his excellent philosophical geek-heroic science fiction adventure (yes, really) Anathem. By no coincidence, Stephenson provides the introduction to the 2010 edition of the present opus.

E&M (to be a little Wallaceish about it) is their love child. If and only if you can read that sentence and think "Hmm ... what a great idea!", this book is for you.