Book Review

The Universe in Zero WordsandIn Pursuit of the Unknown

Reviewed by Gerald B. Folland

The Universe in Zero Words: The Story ofMathematics as Told through EquationsDana MackenziePrinceton University Press, April 2012224 pages, US$27.95ISBN-13: 978-0691152829

In Pursuit of the Unknown: 17 Equations ThatChanged the WorldIan StewartBasic Books, March 2012352 pages, US$26.99ISBN-13:978-0465029730

A decade ago, Graham Farmelo got a dozenscientists and science writers to contribute essayson the “great equations of modern science” andcompiled them into an interesting and informativebook titled It Must Be Beautiful [2]. That book hasnow, it seems, engendered twin children. This year,two well-known expositors of mathematics, DanaMackenzie and Ian Stewart, have simultaneouslyand independently produced books that employ alist of “great equations” (24 of them for Mackenzie,17 for Stewart) as a framing device for discussingmathematics and its impact on civilization. Dowe have a case of overkill here? Perhaps. But let

Gerald B. Folland is emeritus professor of mathematicsat the University of Washington. His email address isfolland@math.washington.edu.

DOI: http://dx.doi.org/10.1090/noti924

me address that question after examining theevidence.

The first point to be made is that, in spite of theircommon theme, the books are quite dissimilar.Farmelo’s book has already been reviewed inthe Notices by Bill Faris [1], so I will say littleabout it here except to note a few points ofcomparison. Most importantly, the fact that it isthe work of multiple authors, only one of whom is amathematician, gives it a decidedly different flavorfrom the books of Mackenzie and Stewart underreview, and its restriction to twentieth-centuryscience gives it a narrower scope. Stewart does notassume much mathematical background on thepart of his readers, and he tries to be very carefulabout explaining the meaning of the ingredientsin his equations. In one of the early chapters heexplains the concept of derivative, and thereafterhe is willing to use it frequently so that somedifferential equations can be put on his list, butthat is about as high as the mathematics goes. (Evenintegrals are described only briefly and used in onlyone chapter.) Mackenzie operates at a somewhathigher level of mathematical sophistication. Hestarts out simply enough—his first equation is“1+ 1 = 2”, which leads into a discussion of thedevelopment of arithmetic in ancient times—butin the later chapters he is not afraid to includesome equations whose precise meaning will beover the heads of many of his readers, as longas he can say something interesting about theirgeneral significance on a nontechnical level.

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Now, what sorts of equations have the authorsfound worthy of inclusion in these “Top n” lists?

To begin with, the winner of the popularitycontest—the one equation that is featured inall three books—is of little intrinsic interest inmathematics and only a small part of a biggerpicture in physics, but it is so famous that it couldnot be omitted. Can you guess what it is? I’ll giveyou a hint: one would be wrong, but not far wrong,to guess that the E on its left-hand side is in honorof Einstein.

The other equations common to both Stewartand Mackenzie have a lot more mathematicalmeat attached to them: the Pythagorean theorem,Newton’s law of gravity, Maxwell’s equations, andthe Black-Scholes equation. (The last of these, asthe concluding topic of both books and one of thetechnical ideas that contributed to the recent globaleconomic crisis, functions as a memento mori : anadmonition that the graces of mathematics can turninto disgraces as well.) There are also other chapterswith a parallel purpose in the two books, althoughthe featured equations are different. For example,both books have a chapter on chaotic dynamics,but Stewart introduces it with the discrete logisticequation xn+1 = kxn(1− xn), whereas Mackenzieuses the Lorenz differential equations with their“strange attractor” solutions. And they each have achapter on quantum mechanics: Stewart choosesSchrödinger’s equation to frame it; Mackenziechooses Dirac’s. (Farmelo has both.)

As for the rest, one will find most of theusual suspects either in the chapter headings orelsewhere in the texts, but the authors have had tostretch things a bit to fulfill their purposes. Five ofStewart’s seventeen “equations” are not actuallyequations, at least if an equation is an assertionthat two apparently distinct things are equal. Threeof them are just the definitions of the derivative,the normal distribution, and the Fourier transform,and one is just the defining relation i2 = −1; theother ringer is an inequality, dS ≥ 0. (S stands forentropy, in case you were wondering.) Similarly, oneof Mackenzie’s equations is π = 3.1415926535 . . . ,and while π is a worthy topic of discussion, itsdecimal expansion is hardly its most interestingfeature. In all these cases as well as others itquickly becomes clear that the equation that headsa chapter is merely what Alfred Hitchcock called“the McGuffin”: the device that sets the plot inmotion.

And most of those plots are quite engaging.For example, Stewart’s chapter on the Pythagoreantheorem begins with some remarks on its emer-gence in the geometry of ancient Greece andelsewhere, then leads the reader through a bit oftrigonometry to Cartesian geometry, nonEuclideangeometry, and finally the apotheosis of the

Pythagorean theorem in itsinfinitesimal form as thecornerstone of Riemann-ian geometry. Mackenzie’smost arcane offering isthe Chern-Weil-Allendoerfer-Gauss-Bonnet formula, whichleads into an account of theeventful life of S. S. Chern andhis role in the developmentof modern differential geom-etry and its connections toquantum physics.

Stewart is writing for thegeneral literate reader. He trieshard to keep the technicalities at a minimum, andhe spends a lot of time discussing historicaldevelopments and the phenomena of the “realworld”, whose study involves the mathematicsin his equations. As a mathematician I some-times wished that he would get to the pointmore quickly, but his discursive style is probablyquite agreeable to his intended audience. As onewould expect from such an experienced writer,he is generally successful in getting the essentialpoints across in a nontechnical way withouttoo much distortion. A few sentences here andthere are obscure, and one equation (the Einsteingravitational equation including the cosmologicalconstant, on page 236) is seriously garbled. Theone chapter that I think quite unsatisfactory is theone on the second law of thermodynamics; I foundStewart’s description of the relation between heatand temperature puzzling and off-kilter, and hisexplanation of situations where the second lawapparently fails is inadequate.

Stewart’s chapter onlogarithms (equation:log xy = log x + log y) fo-cuses on their originalemployment as a (power-ful!) labor-saving devicefor numerical calculations.But computers have nowrendered that use obsolete,and Stewart is at a bit ofa loss to come up withmodern applications. Thechapter ends rather lamely bypointing out that the decibelscale of sonic intensity is alogarithmic one and that tofind the half-life of an isotope from its decaylaw N = N0e−kt , one needs to know what log 2is. I think Stewart has missed a trick by notpicking up on the great structural feature of hislogarithmic equation: it is, to use a professionalterm that he would probably not want to employ,

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one of the earliest examples of a group homomor-phism. With a little work, he could have used itto introduce one of the great themes of modernmathematics—the exploitation of correspondencesbetween apparently different structures and someof its many recent applications.

Mackenzie’s ideal readers have a little moremathematical background than Stewart’s; I envi-sion them as bright undergraduates or high schoolseniors who are oriented toward mathematicsand science or people to whom this descriptionwould have applied in the not-too-distant past.They should probably know some calculus, andthey should be comfortable enough with symbolicexpressions to be able to look at an unfamiliarone with more curiosity than distaste. Mackenzie’schapters are pithier and generally more mathemat-ically adventurous than Stewart’s (and therefore,for my taste, more fun to read). For the ideal readerjust described, they should serve as inviting door-ways into many intriguing areas of mathematics.Like Stewart, Mackenzie is skillful at sketchingthings with clarity and simplicity. His weak spot isthe chapter on the Dirac equation, which containsseveral confusions and misstatements.

In summary, the books of Mackenzie and Stew-art, as well as Farmelo, are all worthy additionsto the popular scientific literature, and they areof sufficiently diverse character that their consid-erable overlap is not mere duplication. However,individually and collectively, they do demonstratethat the “great equations” conceit is not particu-larly natural or productive and that the attempt toshoehorn a wide range of mathematics into thisformat is a procrustean one. I don’t think we nowhave a surfeit of “great equations” books, but wedo have a sufficiency.

References[1] W. G. Faris, Review of It Must Be Beautiful: Great Equa-

tions of Modern Science, by G. Farmelo, Notices Amer.Math. Soc. 50 (2003), 361–367.

[2] G. Farmelo (ed.), It Must Be Beautiful: Great Equationsof Modern Science, Granta Books, London and New York,2002.

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