Joan L. Richards

The art and the science of British algebra: A study in the perception of mathematical truth

Abstract This paper investigates the origins of the concept of mathematical truth by focusing on the development of algebra in England in the early 19th century. In particular, it investigates the reasons why the English, despite their attention to the elements of abstract algebra, never produced a system comparable to modern algebra. Special consideration is given to the works of George Peacock, Augustus DeMorgan, William Whewell, and John Herschel. It is argued that what separated the early development of English algebra from modern algebra is a fundamental difference between 19th- and 20th-century views of truth.

God, Truth, and Mathematics in Nineteenth Century England

Abstract In his Essay Concerning Human Understanding, John Locke created a special epistemological category for mathematical and religious knowing. This category of knowledge was quickly brushed to the side in the French Enlightenment, but the English preserved it well into the nineteenth century. This article considers the ways that the neo-Lockian joining of mathematics and theology fundamentally affected both mathematical and theological thinking in the first half of the English nineteenth century. It argues that these developments set the stage for the post-Darwinian conflicts between science—including mathematics—and religion.

Introduction: Fragmented Lives

Sophia De Morgan’s Memoir of Augustus De Morgan highlights the difficulty of creating a unified picture of a scientific life. It also provides a critical perspective from which to view the chronological development of the modern “scientist” from the eighteenth through the twentieth centuries.

Historical Mathematics in the French Eighteenth Century

At least since the seventeenth century, the strange combination of epistemological certainty and ontological power that characterizes mathematics has made it a major focus of philosophical, social, and cultural negotiation. In the eighteenth century, all of these factors were at play as mathematical thinkers struggled to assimilate and extend the analysis they had inherited from the seventeenth century. A combination of educational convictions and historical assumptions supported a humanistic mathematics essentially defined by its flexibility and breadth. This mathematics was an expression of l’esprit humain, which was unfolding in a progressive historical narrative. The French Revolution dramatically altered the historical and educational landscapes that had supported this eighteenth‐century approach, and within thirty years Augustin Louis Cauchy had radically reconceptualized and restructured mathematics to be rigorous rather than narrative.

De Morgan's logic

This chapter describes three of De Morgans chief contributions, and explores the ways he created a purely symbolic and instrumental notation, taking his cues from developments in symbolic algebra, but inventing a completely independent symbolic system. The chapter also considers the ways he used his notation to rework and extend the world of Aristotelian syllogisms, which led to new interpretations of contraries and contradictories. The chapter explores the way, in one of his last papers, how he expanded his work to embed Aristotles work into a more expansive logic of relations. In his logic, the existential charge, so to speak, stood equally as the conceptual pointer from the laws of thought to a parallel, real universe beyond. The conceptual core of De Morgans deep conservatism harbored the belief that everything humans conceive, think, or imagine in the universe of their logic, sensations, or psychology takes them outside themselves, into another, parallel universe, that “natural syllogism” of the outside world.

“This Compendious Language”: Mathematics in the World of Augustus De Morgan

Mathematics is the most chameleon of subjects, whose meaning is differently defined in different circumstances. This essay considers the mathematics of Augustus De Morgan as an illustration of the ways that the essence of the subject, the very objects that are included within it, has been adjusted in response to cultural factors. Since these cultural factors are the same ones that shape scientific development, the argument is that the history of mathematics and the history of science are always inextricably bound to one another.

In Search of the “Sea-Something”: Reason and Transcendence in the Frend/De Morgan Family

Argument This paper traces the changing fortunes of natural theology in two generations of an English family. The group is represented in the first generation by the Unitarian radical, William Frend, and in the second by the spiritualist Sophia Frend De Morgan and her husband, the mathematician Augustus De Morgan. The Frend/DeMorgans were distinguished from the naturalistic Darwins by their commitment to reason; they were a quintessentially urban group whose impulses to natural theology flowed from a God they encountered through their minds. For William Frend, natural theology was rooted in a tightly disciplined, linguistic view of reason that linked theological and mathematical understandings; in the De Morgans generation, however, understandings of mathematics and of language changed in ways that undercut the power of Frends reasoned approach. In its rise and fall this family story of reasoned theology is co-temporal with the natural theology that was wrecked on the rocks of Darwinian evolutionary theory.

Eloge: Erwin Hiebert.

Erwin Hiebert died on 28 November, 2012, less than three months after his wife Elfrieda died, on 2 September. For the dozens of graduate students, colleagues and friends who tasted the intellectual excitement of discussions in the warmth of their home, these deaths mark the passing of an age. For more than three decades, as a Professor of the History of Science first at the University of Wisconsin, Madison, then at Harvard, Erwin passionately engaged his students and colleagues in explorations that ranged from nuclear physics to experimental physiology, thermodynamics to Cantorian set theory, and quantum mechanics to comets. Acoustics was always a favorite subject, and the best evenings were those that Elfrieda closed by playing on her beloved piano. Erwin was the third of seven children of a Mennonite Bretheren minister, who raised his children in an urban Russian Mennonite community in Winnipeg, Manitoba. In later years, he loved to point to his early education, first in Faraday Grade School and then in Sir Isaac Newton High School, as indicators of his subsequent career as a Historian of Science. However, that career was far in the future for a young man who spent his summers following the wheat harvest from Oklahoma to the Dakotas to pay for his post-secondary education at Tabor College in Hillsboro, Kansas. After two years he transferred to Bethel College, where in 1941 he earned a bachelor’s degree in Mathematics and Chemistry. In 1943 he received a Master’s degree in Chemistry and Physics from the University of Kansas in Lawrence. Also in 1943, Erwin married Elfrieda, née Franz. Elfrieda was already a highly accomplished pianist, who in 1938 had received the highest award in the National Music Competition in Colorado Springs, Colorado; when the two met, she was studying music at Tabor College. Immediately after their marriage, the young couple moved to Chicago, where in 1945 Elfrieda earned a Bachelor’s Degree and in 1946 a Master’s degree in Music from the University of Chicago. Erwin was enlisted as a Research Chemist at Standard Oil Company of Indiana in those years, and Elfrieda was Assistant Music Librarian for the University of Chicago. Erwin’s work with Standard Oil was under the jurisdiction of the Chicago Metallurgical Labs of the Manhattan Project; “within months of the Japanese surrender in August 1945” he and other scientists were coming together “to discuss the political and social responsibility of the scientist, Erwin and Elfrieda Hiebert at their Sixtieth Wedding Anniversary Celebration at Longfellow’s Wayside Inn, Sudbury, Mass., in Summer, 2003. Courtesy of the Hieberts’ children.

The Character of Truth

I was first introduced to Sam Schweber in Harvard’s history of science department, in the 1970s. At the time I was deep in a study of Victorian mathematics and Schweber was captivated by John Herschel.1 Soon thereafter Schweber shifted his gaze to the twentieth century world he knows so uniquely well, but I remained in Victorian England, where Herschel still speaks to me with Schweber’s quiet, gentle voice. One of the central things that he, that is Herschel, says is: “The grand and indeed only character of truth is its capability of enduring the test of universal experience, and coming unchanged out of every possible form of fair discussion.”2 Herschel offered this definition of truth to fend off a certain kind of defensive religious view of science, but the writing that surrounds his description shows that for him scientific truth was at best tenuously distinguished from the religious “truth that sets us free”. Herschel’s views of scientific and religious truth-seeking flowed seamlessly into one another as in his description of natural philosophers as “the best and noblest benefactors of our species” whose work has “raised them above their fellow mortals, and brought them nearer to their Creator.”3 In the 1970s and 1980s, when Schweber and I were traveling together in Victorian England, Susan Cannon was recognizing this view of truth as an essential aspect of the world of the early Victorians; the way Cannon put it was that “for cultured early Victorians, natural science provided a norm of truth.”4 The defining feature of this Victorian view of truth was its unity, the conviction that all that was scientifically or religiously valid was encompassed by a single truth. Cannon saw this unitary view of truth as having been “shattered” after 1859, when “moralists and theologians” refused