17 equations that changed the World (#1)

I do not know if it is a coincidence or if publishers were competing for the same audience: after reviewing The universe in zero word: The story of mathematics as told through equations, in this post (and in CHANCE, to appear in 25(3)!), I noticed Ian Stewart’s 17 equations That Changed the World, published in 2011, and I bought a copy to check the differences between both books.

I am quite glad I did so, as I tremendously enjoyed this book, both for its style and its contents, both entertaining and highly informative. This does not come as a big surprise, given Stewart’s earlier books and their record, however this new selection and discussion of equations is clearly superior to The universe in zero word! Maybe because it goes much further in its mathematical complexity, hence is more likely to appeal to the mathematically inclined (to borrow from my earlier review). For one thing, it does not shy away from inserting mathematical formulae and small proofs into the text, disregarding the risk of cutting many halves of the audience (I know, I know, high powers of (1/2)…!) For another, 17 equations That Changed the World uses the equation under display to extend the presentation much much further than The universe in zero word. It is also much more partisan (in an overall good way) in its interpretations and reflections about the World.

In opposition with The universe in zero word, formulas are well-presented, each character in the formula being explained in layman terms. (Once again, the printer could have used better fonts and the LaTeX word processor.) The (U.K. edition, see tomorrow!) cover is rather ugly, though, when compared with the beautiful cover of The universe in zero word. But this is a minor quibble! Overall, it makes for an enjoyable, serious and thought-provoking read that I once again undertook mostly in transports (planes and métros).

“Riemann, a brilliant mathematical talent (…), in a rush of blood to the brain, he also suggested “On the hypotheses which lie at the foundation of geometry”. Gauss (…) naturally selected it for Riemann’s examination.” (p.18) 

The first equation in the book is Pythagoras’ Theorem, starting with the (indeed) “terrible pun about the squaw on the hippopotamus” (p.3) not worth repeating here. It contains several proofs of the result, along with the remarkable fact that Babylonians were aware of the result in (circa?!) 7289 BC…! However, beyond historical connections, the chapter soon embarks upon a serious introduction to trigonometry. Then Euclidean, non-Euclidean, and even Riemannian geometries.

The second chapter is about the invention of logarithms, with the fundamental feature of transforming products into sums, their applications in astronomy, the sliding rule (of my high school years!), and the use of logarithms in the prediction of radioactive decay (with a completely superfluous paragraph on the Fukushima disaster!). As in The universe in zero word, the use of e as the basis for the natural logarithms remains unexplained.

“The world view of humanity did not suddenly switch from religious to secular. It still has not done so completely, and probably never will.” (p.38) 

The third chapter is based on the definition of the derivative. Unsurprisingly, the author being English means that Isaac Newton gets the lion’s share in this formula, as well as in several other chapters. (Actually, to be fair, Gauss appears just as often in the book!) I find the above quote highly interesting for the long-term impact calculus had on Newton’s and Leibniz’ contemporaries’ view of the world. By putting mathematical formulas behind the motion of planets, Newton and his predecessors, like Galileo and Descartes, “created the modern world” (p.52). (The quote also reminds me of the atheistic reply of Molière’s Don Juan “Je crois en deux et deux sont quatre, Sganarelle, et que quatre et quatre sont huit“, which is easier to put in a theater play than the formula for the derivative!) This chapter contains explanations on potential energy that are the clearest I ever read.

Chapter 4 centres on Newton’s Law of Gravity, in a sort of logical continuity with the previous chapter. I find this formula of gravitational attraction both fascinating and mysterious, and I have always been intrigued at how Newton (and others) had come with it: the chapter does a very good job of explaining this derivation. It also covers Poincaré’s attack of the three body problem and the use of gravitational tubes to significantly improve the efficiency of space travel. (With his mistake about chaos, at the core of a conference given by Cédric Villani in several French universities this year.)

“Apparently, you can thread a hole through another hole, which is actually a hole in a third hole. This way a lies madness.” (p.95) 

The next chapter and its equation i²=-1 are more traditional (and intersect with the list of The universe in zero word), even though Stewart covers series expansions and complex exponential therein. Chapter 6 briefly dabbles in topology, thanks to Euler’s formula for polyhedra and its generalisation. All the way through to knot theory, with Alexander’s and Jones’ polynomials.

“Fisher described his method as a comparison between two distinct hypotheses: the hypothesis that the data is significant at the stated level, and the so-called null hypothesis that the result are due to chance.” (p.122) 

The only chapter truly connected with statistics is Chapter 7 providing the normal density as its equation. It is not the best chapter of the book in my opinion as it contains a few imprecisions and hasty generalisations, the first one being to define the density as a probability of an event on the front page and again later (p.115). And to have the normal distribution supposed to apply in a large number of cases (p.107), even though this is moderated in the following pages. What is interesting though is the drift from Pascal, Gauss, Legendre, and Laplace towards Quetelet, Galton, and then Fisher, the Pearsons, and Neyman, missing Gosset. Professor Stewart mentions Galton’s involvment in eugenics, but omits the similar involvment of Fisher and Pearson, as editors of the Annals of Eugenics. The chapter confuses to some extent probability and statistics, calling the stabilisation of the frequencies a “statistical pattern” when it simply is the Law of Large Numbers. The above quote reflects some of the imprecision transmitted by the author about testing: the fact that “the data is significant at the stated level” α means that it does not support the null hypothesis, not that the alternative hypothesis is true. Professor Stewart states as much in the following sentence, stressing that rejecting the null “provides evidence against the null not being significant” (p.123), but I find it unfortunate that the tool (checking that the data is significant at level α, which means checking a certain statistic is larger than a given quantile) is confused that way with the goal (deciding about a parameter being different from zero). He further subsumes his first mistake by asserting that “the default distribution for the null hypothesis is normal: the bell curve”. This unifies the chapter of course and there is some sensible justification for this shortcut, namely that the null hypothesis is the hypothesis for which you need to specify the distribution of the data, but this is nonetheless unfortunate. Rather interestingly, the chapter concludes with a section on another bell curve, namely the highly controversial book by Herrnstein (no typo!) and Murray, arguing for racial differences in the distribution(s) of the IQ. This thesis was deconstructed by statisticians in Devlin et al.’s Intelligence, Genes, and Success, but 17 equations sums up the main theme, namely the huge limitations of the IQ as a measure of intelligence. Neither Bayes’s formula, not Thomas Bayes are to be found in the book, just as in The universe in zero word. (Jeffreys is one of the two Bayesians mentioned in 17 equations That Changed the World, but in his quality of a seismologist, see tomorrow.)

(to be continued…)