17 equations that changed the World (#2)
“If you placed your finger at that point, the two halves of the string would still be able to vibrate in the sin 2x pattern, but not in the sin x one. This explains the Pythagorean discovery that a string half as long produced a note one octave higher.” (p.143)
The following chapters are all about Physics: the wave equation, Fourier’s transform and the heat equation, Navier-Stokes’ equation(s), Maxwell’s equation(s)—as in The universe in zero word—, the second law of thermodynamics, E=mc² (of course!), and Schrödinger’s equation. I won’t go so much into details for those chapters, even though they are remarkably written. For instance, the chapter on waves made me understand the notion of harmonics in a much more intuitive and lasting way than previous readings. (This chapter 8 also mentions the “English mathematician Harold Jeffreys“, while Jeffreys was primarily a geophysicist. And a Bayesian statistician with major impact on the field, his Theory of Probability arguably being the first modern Bayesian book. Interestingly, Jeffreys also was the first one to find approximations to the Schrödinger’s equation, however he is not mentioned in this later chapter.) Chapter 9 mentions the heat equation but is truly about Fourier’s transform which he uses as a tool and later became a universal technique. It also covers Lebesgue’s integration theory, wavelets, and JPEG compression. Chapter 10 on Navier-Stokes’ equation also mentions climate sciences, where it takes a (reasonable) stand. Chapter 11 on Maxwell’s equations is a short introduction to electromagnetism, with radio the obvious illustration. (Maybe not the best chapter in the book.)
“To understand the Carnot cycle it is important to distinguish between heat and temperature (…) In a sense heat is a bit like potential energy.” (p.202)
Chapter 12 discusses thermodynamics, not through the traditional pV=RT, but rather with the second law of the increase of entropy (which is precisely an inequation, as remarked by Stewart in his notes). It explains this elusive notion (entropy) via the Carnot cycle and its transform into a perfect rectangle, one of the most efficient descriptions I have seen on the topic! Brownian motion (to re-enter the scene later!) is mentioned as being a major step in accepting Botlzmann’s kinetic interpretation. (The final discussion about the connections between time-reversibility and entropy gets a bit confusing, esp. as it concentrates on scrambled eggs!)
Chapter 13 is covering the (almost compulsory) E=mc², explaining some of Einstein’s inputs in a rather satisfactory manner, covering the Michelson-Morlay experiment (which we use as a benchmark in the incoming edition of Bayesian Core) making links with Galileo, Maxwell and non-Euclidean geometries, since it introduces the Minkowski space-time representation. It also debunks a few myths about Einstein, and then moves to highly interesting issues like space warped by gravity, the precession of Mercury (with Le Verrier’s “discovery” of Vulcan), and cosmology. It even goes as far as spending a few pages on the current theories offered for modernising the cosmological paradigm. (The chapter also mentions the now resolved Gran Sasso controversy about neutrinos travelling “faster than light”, p.226.)
“The notion of information has escaped from electronic engineering and invaded many areas of science, both as a metaphor and as a technical concept.” (p.281)
I am afraid I will skip describing Chapter 14 about quantum theory as I find it spends too much time on Schrödinger’s cat, which makes it sound like the hype vocabulary so readily adopted by postmodernists, without any understanding of the Physics behind. And botching its explanation of the quantum computer. (Even though I liked Stephen Hawking’s quote of the multiverse formalism as “conditional probabilities“, p.262!) Chapter 15 gets back to entropy, by being centred on Shannon’s information formula. Maybe because I wrote one of my master projects on error-correcting-codes, I was not very excited by this chapter: even though Stewart ends up with DNA and its information content, the impact of Shannon’s definition does not seem of the same scale as, say, Newton’s Law of Gravity. It is also fraught with the same danger as the previous notion, namely to use it inappropriately. Not so coincidentally, Edwin Jaynes makes a natural appearance in this chapter (as an “American physicist“, p. 282, despite having written a Probability Theory dedicated to Harold Jeffreys and being the charismatic father of the maximum entropy principle): Stewart signals that Jaynes stressed the inappropriateness of assimilating entropy with missing information in every context. Chapter 16 deals with another hype theory, namely chaos theory, by picking the logistic chaotic equation (or map)
which exhibits unpredictable patterns while being completely deterministic and that I used to play with when I got my first personnal computer. This chapter does not get much farther, even though it mentions Poincaré (again), as well as Smale, (my former Dauphine colleague) Arnold, and, obviously, Lorenz. While we do not escape the apparently unavoidable butterfly story, we at least avoid getting dragged into the Mandelbrot set, as would have been the case twenty years ago. However, the chapter fails to explain in which sense chaos theory is more than a descriptive device, except for a very short final paragraph.
“How comes the biggest financial train wreck in human history come about? Arguably, one contributor was a mathematical equation.” (p.298)
As in The universe in zero word, the last chapter is about Black and Scholes formula, again maybe inevitably given its presupposed role in “ever more complex financial instruments, the turbulent stock market of the 1990s, the 2008-9 financial crisis, and the ongoing economic slump” (p.295)… Despite its moralising tone, the chapter does a reasonable job of explaining the mechanisms of derivatives, (even though I prefer xkcd approach!) starting with the Dōjima rice future market in the Edo era (actually, I found that the description on page 298-299 parallels rather closely the Wikipedia article on the Dōjima Rice Exchange!). It also covers Bachelier’s precursor work on the (potential) connection between stock markets and the Brownian motion introduced earlier. Opening the gates for blaming the “bell curve” on not predicting “black swans” (with Stewart referencing Taleb’s book on page 301, and mostly rephrasing Taleb’s leading theme about fat tails in the following pages). The following reference is a talk given by Mary Poovey at the International Congress of Mathematicians in Beijing, 2002, pointing out the dangers of the virtual money created by financial markets, especially derivatives. I presume many prophetic warnings of that kind could have served the same purposed, as fulfilled prophecies are rather easily found a posteriori.
“The Black-Scholes equation is also based on the traditional assumptions of classical mathematical economics: perfect information, perfect rationality, market equilibrium, the law of supply and demand (…) Yet they lack empirical support.” (p.310)
What I find most interesting about this last chapter is that it is about a formula that did not “work”, in opposition to the previous sixteen formulas, although it equally impacted the World, if possibly (and hopefully) for a short time. Since it is more or less the conclusive chapter, it has a rather lukewarm feeling of the limitations of mathematical formulas: as bankers blindly trusted this imperfect representation of stock markets, they crashed the World’s economies. This turning the Black and Scholes formula into the main scapegoat sounds a wee simplistic, especially given the more subtle analyses published after the crisis. See also this BBC coverage by Tim Hartford. (I am not even sure that the Black-Scholes equations were ever adopted by their users as a way to represent reality, but rather as a convention for setting up prices.) The universe in zero word was much more cautious (too much cautious) about what caused the crisis and how much the Black-Scholes equation was to blame… Maybe the most popular chapter in the book, to judge from reviews, but rather off-the-mark in my opinion.
Interestingly, the final page of the book (p.320) is a sceptical musing about the grand theories advanced by Stephen Wolfram in A new kind of Science, namely that cellular automata should overtake traditional mathematical equations. Stewart does not “find this argument terribly convincing” and neither do I.