By John P. Boyd
“To turn events into ideas is the function of history.” – George Santayana
Applied mathematics has a history, but who will tell it? SIAM News periodically publishes obituaries of its esteemed members, but the honorees’ accomplishments are communicated only through the brief summaries and humorous anecdotes of their memorialists. Mathematical and scientific concepts evolve and generate new ideas, but events turn into ideas as well. The mix of elaborations and calculations that embody the quirks and contingencies of individuals, culture, wars, fads, and politics makes it arguably impossible to fully understand the present and future without also having some grasp of the past.
If history is idealized as a smooth, continuous flow from plain to baroque, then vector calculus (universally applicable with three components) should have preceded quaternions (few applications, complicated theory, and four components) by half a century. Lord Kelvin noted that “Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.” But the graffiti that William Rowan Hamilton scratched on Broom Bridge on October 16, 1843 was the non-commutative multiplication law of quaternions. Vector calculus was shaped mostly by Josiah Willard Gibbs and Oliver Heaviside four decades later [1].
In the mindset that probability implies inevitability, Gibbs should have perished like 750,000 others of his generation in the American Civil War. But he had chronic lung disease and therefore never enlisted. He should have contracted tuberculosis, which killed his mother, and died by age 25; instead he lived into his 60s — just long enough to teach Edwin Bidwell Wilson. Since Gibbs refused all entreaties to write a book on vectors, his lecture notes of 1883 should have remained the secret doctrine of his small cohort of students. But Wilson, who was only 22 years old, filled the void with Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, Founded Upon the Lectures of J. Willard Gibbs; this was the first book on modern vector theory. Gibbs died unexpectedly of an intestinal blockage less than four years after Wilson took his class at Yale.
One view of mathematics is that it (metaphorically) evolves monotonically from archaea to the 12-primary-color, polarization-resolving eyes of the mantis shrimp. This viewpoint asserts that Jule Charney’s theory of baroclinic instability in the ocean and atmosphere, which was solved by confluent hypergeometric functions, should have appeared at least five years after the simpler so-called “f-plane” theory whose solutions are merely hyperbolic functions. But history is indifferent to the prognostications of the learned; illogically, it was not until five years after Charney that Eric Eady took the Great Leap Backwards by omitting the so-called “beta effect,” thereby showing that it was significant but not essential for baroclinic instability [3].
Meteorology was converted from an empirical science—a sort of botany-of-clouds—to a branch of physics and fluid mechanics around 1900. This happened in large part because Vilhelm Bjerknes put filial piety above his dreams of a brilliant physics career in Germany and returned to Stockholm to help his ailing father finish his life’s work: a massive treatise read by none but its authors. His colleague Nils Ekholm, a pioneer aerodynamicist and survivor of the unsuccessful 1896 balloon expedition to the North Pole, encouraged Bjerknes to delve into weather.1 A new branch of physics was born in an academic backwater from a man who was battered by years of depression and well into middle age.
The most unlikely of Bjerknes’ many allies was neurobiologist Fridtjof Nansen. Though Nansen was unlearned in both mathematics and physical oceanography, he was a tireless and careful observer. Under the supervision of Bjerknes, Vagn Walfrid Ekman turned Nansen’s puzzling but unchallengeable measurements of the Arctic Sea into explanations. And thus Nansen—far better at skiing and persuasion than algebra—was father to the mathematics behind the Ekman current spiral, Ekman boundary layer, Ekman pumping and Ekman suction, and the theory of “dead water.” Tiny, remote Scandinavia dominated atmospheric physics for several decades mostly because Bjerknes happened to be Norwegian [2].
What might Évariste Galois have done if he had not been killed in a duel at age 20? Or Bernhard Riemann, who died of tuberculosis at 40? Contingency blasts holes again and again in the smooth evolution of mathematics. Unfortunately, misunderstanding the roles of contingency and personality is not the only cost of poor or nonexistent mathematical history.
Some organizations have begun making deliberate commitments to preserving the history and biography of women and underrepresented applied mathematicians. But even so, much is lost: Sylvia Skan of the Falker-Skan equation; Anne Nicolson of the Crank-Nicolson algorithm; Grace Vaisey, who earned Sir Richard Southwell his knighthood; Lorna Swain, who became a hydromechanics lecturer at the University of Cambridge in 1926; and Susan Martin, who rose from calculator girl to senior scientific officer over the course of 57 years at the National Physical Laboratory. Does anyone know the names of these pioneers?
Cicely Ridley was a pathfinder in numerical quantum chemistry and earned a doctorate in 1956 under Douglas Hartree. At age 30, she suspended her career to raise four children. Ridley returned to work as a grunt-level programmer/consultant at the National Center for Atmospheric Research (NCAR),2 where she rapidly rose to NCAR’s highest non-administrative rank of senior scientist. She was well known in atmospheric science as a co-creator of the Roble-Dickinson-Ridley computer code: the first general circulation model of the thermosphere. Cicely is a great role model, but she can no longer tell her story.
Many years ago, I was the summer research advisor for an African American undergraduate named Rudy Horne. Decades later as a tenured associate professor at Morehouse College, Horne was invited back to the University of Michigan to deliver a much-anticipated keynote lecture on Martin Luther King Jr. Day in 2018. He had spent four months as the mathematics consultant for the movie Hidden Figures, during which he educated the cast about the joy and practical uses of mathematics. In the film, Katherine Johnson’s solution of a quadratic equation is in his handwriting. Horne could have told amazing stories about his experiences, but he died suddenly at age 49 before delivering the lecture.
History and biography are in a constant race against time to preserve documents, oral histories, and artifacts before death and landfills render them forever inaccessible. In this race to preserve and protect, many societies are competing by standing still. SIAM has superb editorial, production, and marketing staffs, as I know from personal experience. However, it does not have a book series for individual and collective biographies, collected papers of distinguished mathematicians, or applied mathematics history. Organizations like SIAM should establish committees to look after the history and biography of applied mathematics. Such committees would then appoint editors and encourage submissions in the aforementioned areas.
Although a robust publishing program in history and biography would take some time to develop, it is certainly doable. SIAM already has a collection of downloadable oral histories, but only in numerical analysis. Prospective historical committees could solicit and encourage the archiving of oral histories, autobiographies, and other materials in all areas of applied mathematics. Formal and informal partnerships with university archives, research libraries, museums, and science history programs would also be beneficial.
Artifacts are as perishable as their makers. The aircraft carrier CV-6—the most highly decorated ship of World War II—would have made a wonderful floating museum. However, the carrier that survived innumerable bombs and a kamikaze hit was scrapped in 1956, defeated by a far more powerful adversary than the Kidō Butai: historical indifference. With this in mind, societies should aim to publicize relevant museum holdings. Cornell University’s College of Engineering has a marvelous collection of pre-electronic calculating machines, and the Science Museum in London contains a differential analyzer (analog computer) that was built from a child’s Meccano/Erector set. The museum also displays Lord Kelvin’s 1876 ocean tide forecaster; in lieu of today’s silicon integrated circuits, Kelvin’s was a flock of brass gears and spheres that was used operationally for half a century. Furthermore, an example of one of the most powerful species of brass-and-electric-motor computers—the U.S. Torpedo Data Computer Mark IV—is still operational on the U.S. Navy submarine Pampanito, which is now a museum ship in San Francisco Bay. Perhaps mathematics organizations can serve as information pipelines between museums and prospective donors who want their mathematical artifacts to be displayed rather than hidden in a drawer by curators who are baffled by ellipsoids.
The recycling of warship names is a reminder that the past is deeply embedded in the future. The name of the CV-6 was born by six U.S. Navy ships before her and two more (to date) after her, a lineage that will presumably extend into the distant future. The true name of the ship that once stood alone against an empire—the last operational carrier in the Pacific Ocean—is now known in every part of the world thanks to a television voiceover on Star Trek that begins, “These are the voyages of the starship Enterprise . . . her continuing mission . . . to boldly go where no one has gone before.” In the name of historical preservation, where will SIAM and other applied mathematics organizations boldly go?
1 Bjerknes’ first and only physics student, Nils Strindberg, was expected to someday metaphorically hang his own shield in the Hall of Weather Heroes, but Ekholm thought that ballooning needed a younger man and persuaded Strindberg to take his place as a weather observer on the 1897 expedition to the North Pole, which was lost with no survivors.
2 I am grateful for Ridley’s help with ordinary differential equation solvers for my thesis when she was still a programmer.
References
[1] Crowe, M.J. (1967). A history of vector analysis. South Bend, IN: University of Notre Dame Press.
[2] Friedman, R.M. (1989). Appropriating the weather: Vilhelm Bjerknes and the construction of a modern meteorology. Ithaca, NY: Cornell University Press.
[3] Vallis, G.K. (2017). Atmospheric and ocean fluid dynamics: Fundamentals and large-scale circulation. New York, NY: Cambridge University Press.
John P. Boyd is a professor in the Department of Climate and Space Sciences and Engineering at the University of Michigan.