### By David Krakauer and Cristopher Moore

Reuben Hersh — mathematician, philosopher, historian, Quaker, Professor Emeritus at the University of New Mexico, and frequent interlocutor and researcher at SFI, passed away on January 3, 2020.

He was best known as the author with Phil Davis of The Mathematical Experience, published in 1981 and winner of a National Book Award, and the later, What is Mathematics, Really? published in 1997. For many of us, The Mathematical Experience served a role not unlike Douglas Hofstadter's Gödel, Escher, Bach, published in 1979, both of which made the world of formal ideas feel light and playful and deeply existential, infusing domains that had hitherto remained under lock and key behind the walls of discipline and department with a little of that spirit of the Enlightenment, the sensibility of Voltaire, and with his anti-establishment irony, rigor, and democratic zeal.

Having started his career in English literature at Harvard and then interning at the magazine Scientific American, Reuben was inspired to return to New York University where he received his degree in mathematics in 1962. Reuben described his own experience this way, “After a while, I remembered that before becoming an English Lit major in college, I had seriously considered majoring in math. But a deadly dull second semester of calculus convinced me not to take any more math courses… I got some workmen’s compensation for my lost half-thumb, collected unemployment insurance as long as possible by pretending to look for a job, and got admitted as a math grad student at New York University.’’ This is a career path that few could take today, but Reuben considered a rule of life that “if I am scared to do something, then that is what I must go and do.”

Reuben’s work connected partial differential equations with the motions of molecules, showing that particles colliding with each other randomly can be modeled with the continuous tools of diffusion and fluid flow. Reuben soon found himself preoccupied with the nature of mathematics itself, both its historical and philosophical foundations and the reasons for its extraordinary power when applied to questions of the natural and the social sciences. At the age of 50, Reuben turned, in his words, from creating and discovering mathematics to asking what it means to be a mathematician—what we do when we do mathematics. He focused not on the formal questions that occupied much of 20th-century logic and philosophy, but on the lived experience of mathematicians working as part of a community.

Reuben sought to account for two aspects of mathematical experience. On one hand, everyone who does mathematics has the sensation that they are encountering objective truths: numbers, shapes, and so on have their own logic and are not subject to our whims. As he wrote, when one manipulates mathematical objects in one’s mind, “The analogy to grappling with and mastering material objects—for example, a sculptor learning to respect the clay, the metal or the stone—is to me inescapable.” This sensation inspires the Platonist view that mathematical objects are as real as physical ones, and that mathematical truths hold independently of our ability to think about them.  On the other hand, mathematics is a human activity and a product of human thought and culture. It is, to a large extent, a game played with rules we invent. So does mathematics exist outside of us, or inside our own minds?

Reuben proposed a middle way that respects both the reality of mathematics and our experience in creating it. To him, mathematical objects are mental objects, but with a higher degree of coherence than, say, political or aesthetic opinions. Once we conceive of a cube, it has six sides whether we want it to or not. This lets us share our mental models with each other, perceive them together, and — almost always, after enough work together — come to agreement about their properties. He wrote, “A world of ideas exists, created by human beings, existing in their shared consciousness.” Just as some parts of the physical world are lawful and predictable, so are some parts of the world of ideas—and the study of these is what we call mathematics.

Reuben was particularly skeptical of what Imre Lakatos called “foundationalism,” the hunger for a logical underpinning for mathematics that would make it absolutely airtight, free of uncertainty or paradox. He argued that this goal remains unattained, and more radically, that Few still hope or yearn for its attainment.”

One of us (David Krakauer) remembers a particularly intense series of conversations with Reuben inspired by Jean Pierre Melville's film “Army of Shadows” —  a masterpiece from 1969 that provides us with rare insights into the French resistance during the occupation of France spanning 1940 to 1944, the period described by the historian Mark Bloch, as the strange defeat.

The rather unlikely leader of the resistance is an unassuming professor of the philosophy of mathematics at the Sorbonne. Reuben and I became fascinated by the real-world Jean Cavaille who served as the inspiration for one of Melville's leading characters. Cavaille had been declared a public enemy during the Vichy regime and fled to London where he had several clandestine meetings with the exiled  Gen. Charles de Gaulle.

Cavaille was willing to risk his life for his compatriots in defiance of the German occupiers and had spent most of his career working on abstract set theory, issues around the nature of the continuum and the transfinite, and mathematical logic more generally.

It strikes us now that this semi-fictional character spanned the two worlds with which Ruben was most preoccupied: the world of rarefied and abstract mathematical reasoning and the world of human rights, which requires that every thinking person engage according to their abilities with the facts of injustice and inequality. Presumably, some part of each of these worlds explained his attraction to Quakerism.

Reading through some of the books from Reuben's library, one is struck by the reassuring diversity and concerns informing his thought: the works of Chekhov and Dostoevsky, James Joyce's Dubliners, the plays of Brecht, Ibsen, and Strindberg, the expectantly numerous monographs on the history and philosophy of mathematics, the collected poems of Pasternak, Philip Larkin, John Berryman, and Robert Lowell, and the early history of science, focusing on the contributions of Galileo, Hook, and Newton.

And then those books that one hardly knew existed that are the secret ingredients in the formation of the minds of iconoclasts, such as the writings of Karl Marx on the limitations of the differential calculus, Willard Bascom's monograph on waves and beaches, C.V. Boys' treatise on soap bubbles, and Gustav Dorey and Blanchard Jerrold's illustrated pilgrimage to London.

To Reuben, mathematics was not a rigid building which will collapse if there is a crack in its foundation. It is more like a tree with ever-renewing roots, or an eagle whose wings are borne up by its creators’ thoughts. It is not a sterile, formal creation: it is “fallible, corrigible, tentative and evolving, as is every other kind of human knowledge…  it is nothing more or less than the exploration of the mathematical environment, which we create and expand as we explore it.” It is one of the most wonderful things humans do, and we can only do it together. And it was exactly in this spirit of creative partnership that Reuben and his partner Vera John-Steiner wrote their final book on Loving and Hating Mathematics in 2011.

Thank you, Reuben, for a lifetime of inquiry and inspiration.

David Krakauer
President and William H. Miller Professor of Complex Systems
Santa Fe Institute

Cristopher Moore
Professor and Science Board member
Santa Fe Institute