Think Piece

Math as “One of the Great Humanities”: Cassius Keyser, Doctrinal Functions, and the Valuing of American Mathematics

by guest contributor Ellen Abrams

By Ellen Abrams

In the 2020 United States presidential election, one of the major candidates for the Democratic nomination, Andrew Yang, substituted the traditional American flag pin on his lapel for one that simply said “MATH.” Attached to this small pin were big assumptions about what “MATH” is and how it should be valued. Yet, both the meaning and value of mathematics in the United States have been up for debate and hard to pin down throughout the nation’s history. During the Cold War, Americans learned to value mathematics for its role in science and industry. “Math is everywhere,” students were told, and falling behind in the field was seen as a threat to national security. Prior to World War II, however, mathematics was not valued as highly in American society, and mathematicians often expressed concern about the cultural and financial status of their discipline. Like many do now, most researchers at the time pursued “modern,” abstract forms of mathematics. Their attention, in other words, remained focused on internally consistent, well-defined systems that were built and studied solely for the sake of their own development. There was very little, if anything at all, that related such abstract mathematical systems to the industrial or political concerns of modern America. As the president of the Mathematical Association of America warned in 1917, if American mathematicians continued to focus almost exclusively on formal abstractions “we must expect and we shall deserve public disdain and sincere doubt of our value to humanity.” Research funding and job prospects were at stake, as was the relevance of their young community. In response to these concerns, American mathematicians worked to cultivate various forms of prestige tied to differing conceptions of their discipline and its value to American society.

Although most American mathematicians were focused on abstract research in the early twentieth century, they nonetheless sought to bolster the status of their discipline by asserting its ties to science and industry. Columbia mathematician Cassius Jackson Keyser, however, took a notably different approach. While acknowledging the usefulness of mathematics, Keyser instead doubled down on its humanness. He understood this difference to be “much like the difference between one who greets a newborn day because of its glory and one who regards it as a time for doing chores and values its light only as showing the way.” Although mathematics in general could be associated with numeracy and an increasing role for quantification in the sciences, Keyser instead emphasized topics and approaches to the discipline that extended its categorization beyond the sciences. “Nothing is better entitled to rank as one of the great Humanities,” he claimed, “than Mathematics itself.”

While mathematics’ association with the sciences brought with it connotations of Anglo-Saxon, middle-class professionalism, mathematics’ affiliation with the humanities reinscribed a different, though similarly exclusionary, image of the discipline tied to celebrations of whiteness and elitism. Both ways of valuing mathematics spoke to either side of an assumed split between “thinking” and “doing.” In January 1924, Keyser addressed the Bureau of Personnel Administration in New York as part of a symposium on “Linking Science with Industry.” His goal was to garner support among an audience of businessmen for programs like the American Mathematical Society’s endowment campaign, which was designed to raise money but also to educate the public “concerning the basic character of mathematics in our present civilization and the importance of mathematical research in advancing that civilization.”[1] While for many “advancing civilization” meant advancing science and industry, to Keyser it meant advancing the cultural ideals of a civilized people and what he later called “the development of humanity’s manhood.” In his speech, Keyser argued that leaders in industry had a duty to support art and “pure” science, regardless of their industrial utility. The purpose of industry, which was not self-justified, was to allow communities of humans to “live a good life” through artistic and intellectual pursuits, which were self-justified.[2] By characterizing mathematics as a “civilized,” humanist pursuit, Keyser reinscribed exclusionary images of a supposedly inclusive discipline, constricting what it meant and whom it was for.

One of the primary ways Keyser worked to align mathematics with humanism and other self-justified pursuits was through the concept of “postulationalism.” In Mathematical Philosophy: A Study of Fate and Freedom; Lectures for Educated Laymen, he began by explaining that each branch of mathematics comprised a set of propositions that were either taken for granted and left unproven, called postulates, or proven through deduction from the others. Many have encountered this distinction in high school geometry class, where students are given a set of statements, like “two distinct points determine a line,” and are told to combine and manipulate them in order to derive various truths about lines and angles. Euclid of Alexandria famously formalized this type of reasoning and applied it to mathematics in the 3rd century BC. Yet, other systems, including “non-Euclidean” geometries, followed in the nineteenth century and were reformulated in David Hilbert’s 1899 Grundlagen der Geometry, or “Foundations of Geometry.” After the publication of Hilbert’s Grundlagen, which defined a new system of axioms, or postulates, for geometry, postulate systems became a distinct topic of interest among mathematicians in the United States. By studying the axioms of different fields of mathematics and fine-tuning different systems, American mathematicians helped to inaugurate a shift from Hilbert’s geometry, or “science of space,” to postulate systems of inherent interest. They focused, for example, on establishing generalizable features of postulate systems, such as consistency (no postulate contradicted any other) and independence (no postulate could be derived from any other). Keyser would eventually define “pure” mathematics as “postulationalism,” or the study of postulate systems themselves as a form of reasoning, separate from any application or content.

To define mathematics as postulationalism, Keyser relied on a concept he referred to as “doctrinal functions.” Hilbert’s Euclidean Geometry, for example, was best defined as a doctrinal function, meaning a system of propositional functions. Propositional functions were similar to functions (statements about the dependence of one or more things on something else) but with one or more undetermined variables to which different meanings could be assigned, such as sin x = cos y. A propositional function was neither true nor false but became true or false when specific constants were assigned to its variables. Although Hilbert’s postulates for geometry may have seemed like propositions with specific content, for example, they were actually propositional functions in disguise. Instead of “any three points not in the same straight line determine a plane,” Hilbert could just as easily have written, “Any three loigs not in the same boig determine a ploig.” In doing so, there would have been no change in the relationship between the postulates (propositional functions that were neither true nor false) and the theorems deduced from them (also neither true nor false). If the variables in a doctrinal function’s propositional functions were, however, replaced by constants, then the propositional functions would become propositions, and the system of propositions would become a doctrine. If the constants happened to include undefined concepts like points, lines, and planes, then the doctrine would be Euclidean Geometry. Defining variables as constants was considered an “interpretation” of the function, and, as Keyser pointed out, every doctrinal function admits of an infinite number of interpretations. Although E.H. Moore, a professor at the University of Chicago and one of the leading figures in American mathematics, also claimed to have established a similar concept to doctrinal functions, Keyser’s supporters credited him with the term.

Other American mathematicians (including Norbert Wiener, R. D. Carmichael, and Keyser’s own student, E. T. Bell) also wrote about mathematics as postulationalism; yet none extended the idea as much or in the same, particularly humanist, directions as Keyser. Even if they did not become full-on doctrinal functions, Keyser insisted that all fields of thought relied on starting principles—i.e., postulates. When saying, “We hold these truths to be self-evident,” the authors of the Declaration of Independence were saying “we lay down the following postulates.” Keyser also claimed there were foundational assumptions at the basis of Darwin’s On the Origin of Species, Marxian socialism, and Einstein’s theories of relativity. Like Euclidean Geometry, these doctrines applied specific content to the general form of postulation.

In many ways, Keyser’s attempts to define mathematics through doctrinal functions and to promote its ties to humanism were well situated within contemporary intellectual concerns and critiques of modern society. For some (notably white, upper-class) Americans, the social and cultural changes of the late-nineteenth and early-twentieth centuries were in need of counteraction. Many feared that increasing urbanization, mechanization, and modernization would mean a loss of traditional values, a decline in intellectual culture, and a weakening of the human (specifically, Anglo-Saxon) race. Fewer men were craftsmen or landowners, and increasing numbers of women were diagnosed with a general weakness and nervousness condition (known as “neurasthenia”) that perpetuated conceptions of female fragility. The stress and chaos of modern life were thought to be too much for women and, by extension, were a threat to the continuance and continued dominance of the white race.

In Mathematical Philosophy, Keyser noted that a genuine understanding of postulates systems “cannot be gained by any of the get-rich-quick methods characteristic of our industrial and neurasthenic age.” In such critiques of modern society, Keyser both clashed and aligned with various strands of anti-modernism. His promotion of mathematics as relevant (often via his concept of doctrinal functions) to every possible human endeavor, for example, was at odds with a more conservative “New Humanist” movement that blamed mathematical reasoning for the corruption of earlier, purer forms of humanism. Other strands of anti-modernism, however, included certain forms of mathematics in its celebration of classical disciplines. Both versions of humanism involved a longing for perceived forms of cultural superiority. Both were imbued with whiteness and elitism.

In general, Keyser’s touting of mathematics as a self-justified, humanist pursuit held to one side of an ongoing struggle between enriching privileged lives and providing basic needs. Financial depressions, widespread poverty, and inhumane working conditions in the United States had spurred ongoing calls around the turn of the twentieth century for modern American leadership to justify its actions. As the retiring president of the Mathematical Association of America, Dunham Jackson, explained, “When the gentleman of leisure was the ideal of society, a gentleman whose activities were harmless was a good citizen. But this democracy which has happened so unexpectedly demands that you do something useful, or at least make out a case for the usefulness of what you do.”

Amid currents of anti-intellectualism, populist politics, and the needs of industrial capitalism, support for American mathematics in the early-twentieth-century was often defined by its usefulness. Nonetheless, ties between the image of the mathematician and the gentleman of leisure remained, in part through humanist commitments like Keyser’s. Both ways of valuing mathematics assumed a particularly white, male professional identity: one a businessman industrialist, the other a gentleman scholar. Keyser’s efforts to define mathematics as an inherently valuable human endeavor were necessarily refracted through his own conceptions of what it meant to be “valuable” and what it meant to be “human.” These conceptions, in turn, helped to shape the professional identity and cultural value of American “MATH” for decades to come.

[1] Council Minutes January 1, 1926, Box 12, Folder 91, American Mathematical Society Records, The John Hay Library, Brown University.

[2] Cassius Jackson Keyser, stenographic report of “Man and Men” presented January 4, 1924, Columbia University Rare Book and Manuscript Collection, Cassius Jackson Keyser papers, Box 9.

Ellen Abrams is a postdoctoral fellow at McGill University and a visiting scholar at Cornell Tech. Her research spans the history of mathematics, data, and organizing.

Featured Image: Cassius Keyser. Scripta Mathematica vol 5, no. 2 (1938).

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