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Celebrating Mathematical Greats

By James Case

Significant Figures: The Lives and Work of Great Mathematicians. By Ian Stewart. Basic Books, New York, NY, September 2017. 328 pages. $28.00.

Significant Figures: The Lives and Work of Great Mathematicians. By Ian Stewart. Courtesy of Basic Books.

Ian Stewart’s latest venture into the realm of popular mathematics should meet, for many years to come, the need for an update of E.T. Bell’s 1937 classic Men of Mathematics. Both books offer chaptered accounts describing the lives, times, and work of one or more great mathematicians. For obvious reasons, neither volume can include all who fit that description. Bell, for example, never speaks of Camille Jordan, and mentions Pafnuty Chebyshev and Siméon-Denis Poisson only in passing. Stewart says nothing of Hermann Minkowski, George David Birkhoff, or Norbert Wiener, and devotes no chapter to John von Neumann or Alexander Grothendieck. Both authors confine their attention to mathematicians deceased at press time, which explains Bell’s neglect of David Hilbert along with Stewart’s of (say) Stephen Smale, Andrew Wiles, and Grigori Perelman.

After brief introductions, both authors commence—naturally enough—with chapters on Archimedes. But Stewart follows with accounts of three other ancients (Liu Hui, Muhammad ibn Musa al-Khwarizmi, and Madhava of Sangamagrama) whose works were almost certainly unknown to Bell, and another on Gerolamo Cardano, whose formulas Bell mentions without attribution. The latter serves as Stewart’s segue into the Renaissance/Enlightenment revival of scholarship, which he covers in significantly less detail than Bell. Following an obligatory chapter on Évariste Galois, Stewart introduces the first of three female mathematicians: Augusta Ada King, Countess of Lovelace. Why King, rather than Maria Gaetana Agnesi, whose book on differential calculus contains an early discussion of the curve known as “the witch of Agnesi;” Sophie Germain, whose contributions to number theory and elasticity were of lasting significance; or perhaps the ancient Hypatia of Alexandria?

King’s claim to fame rests on her collaboration with Charles Babbage, inventor of the difference and analytical engines. She translated into English the published version of notes taken by Luigi Federico Menabrea at one of Babbage’s lectures on the analytical engine. At Babbage’s suggestion, King added commentary of her own, until the comments quite eclipsed the notes.

Her comments illustrated the machine’s capabilities with a series of examples, the most ambitious of which involved the Bernoulli numbers, best known as the coefficients of \(x^n/n!\) in the Taylor series expansion of \(x/(1-e^{-x})\). They explained, at least in principle, how one could program the machine with punched cards—of the sort employed in Joseph Jacquard’s then-celebrated loom—to function as a universal computer capable of carrying out any imaginable computation. Though one historian finds “not a scrap of evidence that Ada ever attempted original mathematical work,” none can deny her importance as Babbage’s interpreter and publicist. In later life, she partook excessively of wine, indulged in opium, had numerous lovers, and left gambling debts in excess of £2,000 upon her death from cancer at age 37. Yet nothing she ever did could dim Babbage’s fascination with the woman he called “the Enchantress of Numbers.”

Stewart’s second leading lady of mathematics is Sofia Kovalevskaya. His account begins when Kovalevskaya turned up in Karl Weierstrass’ Berlin office, hoping to become his pupil. She quickly astonished him with the originality of her solutions to problems he assigned, including a demonstration that solutions of the initial value problem for the backward heat equation are not unique. She possessed, as he later put it, “the gift of intuitive genius.”

Although she published only 10 mathematical papers in her lifetime, Kovalevskaya’s discoveries amazed the leading figures of her day. In partial differential equations, mechanics, and the diffraction of light by crystals, her work was penetrating, original, and technically proficient. She was appointed professor ordinarius at Stockholm University in 1889—a paid, tenured position—becoming the first woman to hold such a post in a northern European university. Soon thereafter, she was elected to occupy a chair in the Russian Academy of Sciences, another first. Kovalevskaya was arguably the foremost female scientist of her generation, eclipsed only by Marie Curie some two decades later. She died of influenza in 1891, less than a month after her 41st birthday.

The third of Stewart’s trio of great female mathematicians is Amalie Emmy Noether. Born to a prosperous Jewish family in the sleepy Bavarian city of Erlangen, she was drawn to mathematics by her father, an accomplished geometer in his own right. Noether was indeed fortunate that her hometown university changed its rules in 1904, allowing women to matriculate on the same basis as men. Having attended lectures by her father and a few others for several years, she was able to obtain her degree that very year and decamp for Göttingen and the study of invariant theory under the eminent Paul Gordan. Though she was awarded her Ph.D. summa cum laude in 1907, Noether found herself ineligible to even apply for a teaching position at a German university, since women were not allowed to pursue habilitation. She therefore returned to Erlangen, where she helped her father with his courses and continued her own research.  Her work paid off, and after seven years David Hilbert and Felix Klein summoned her back to Göttingen, then the center of the mathematical universe.

Noether is perhaps best known for a theorem she proved soon after returning to Göttingen, associating conservation laws with symmetries in the laws of nature: Time translation symmetry yields conservation of energy, space translation symmetry yields conservation of momentum, rotational symmetry yields conservation of angular momentum, and so on. Albert Einstein praised the theorem—proven in 1915—as a piece of “penetrating mathematical thinking.” It has since become a standard meta-theorem of mathematical physics. In its original version, the theorem applies to Lagrangian motion.

Consider a particle moving on a line with Lagrangian \(L(q, \dot{q})\), where \(q\) is its position on the line and \(\dot{q}\) is the velocity. The momentum of the particle is \(p=\partial L/\partial \dot{q}\), while the applied force is \(F=\partial L/\partial \dot{q}\). The so-called Euler-Lagrange equation then asserts that the rate of change of momentum \(\dot{p}\) equals the force \(F\).

Next, suppose the Lagrangian \(L\) to be invariant under a one-parameter family \(\{T_s\}\) of symmetries, so that \(L(T_sq, T_s\dot{q})\) remains constant. Then \(C=p \cdot d(T_sq)/ds\) is a conserved quantity, since \(\dot{C}\) vanishes identically. It does so because

\[\dot{C}=\dot{p} \cdot \frac {d(T_sq)}{ds}+p \cdot \frac {d(T_s\dot{q})}{ds},\]

which becomes

\[\dot{C}=\frac{\partial L}{\partial q} \cdot \frac {d(T_sq)}{ds}+ \frac{\partial L}{\partial\dot{q}} \cdot \frac{d(T_s\dot{q})}{ds}=\frac{dL(T_sq,T_s\dot{q})}{ds}\equiv 0\]

upon substitution of \(\partial L / \partial \dot{q}\) for \(p\), \(\partial L / \partial q\) for \(\dot{p}\), and appeal to the chain rule. 

There is, of course, nothing special about a particle traveling on a line. One can easily generalize the proof (via subscripts) to collections of particles in higher-dimensional space, or even on manifolds. However, it only works if the symmetries in the group \(\{T_s\}\) are time-dependent, so that

\[\frac{d}{dt} \left[\frac{d(T_sq)}{ds}\right]=\frac{d}{ds}[T_s\dot{q}].\]

More complicated versions of Noether’s theorem are required to handle groups of time-dependent symmetries, such as the Lorentz transformations of special relativity.

In 1919, the University of Göttingen finally capitulated to Hilbert’s nagging pressure, first approving Noether’s habilitation and then conferring the rank of Privatdozent. Soon thereafter she changed fields, taking up the study of abstract algebra and bringing it—in collaboration with Bartel Leendert van der Waerden, among others—to roughly the condition described in his 1931 text Modern Algebra. Noether later developed an interest in topology, becoming a founder of algebraic topology.

She remained quite happily at Göttingen until 1933, when the Nazis dismissed all Jewish people from university positions and obliged her to move to Bryn Mawr College near Philadelphia, Pa. The new location was admirably convenient to the then-new Institute for Advanced Study in Princeton, N.J., where Noether was regularly invited to speak. She died in 1935, of complications following a cancer operation.

David Hilbert, Kurt Gödel, Alan Turing, Benoit Mandelbrot, and William Thurston are among the mathematicians who lived too long for inclusion in Bell’s volume. Stewart devotes a chapter to each. Though Srinivasa Ramanujan died in 1920, his work was not yet sufficiently well-enough known (or widely-enough understood) in 1937 to justify inclusion by Bell. Stewart rectifies the omission.

Because nearly all the aforementioned mathematicians have been the subjects of recent biographies, there seems no need to recap Stewart’s summaries here. Suffice it to say that Stewart has written a worthy successor to Bell’s far-from-outdated classic — one that may in time incline an even greater number of young readers to pursue careers in mathematics. Meanwhile, working professionals curious about the lesser-known masters profiled in the book, yet lacking the time or inclination to digest an entire biography, will find Significant Figures both informative and entertaining.

James Case writes from Baltimore, Maryland.

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