## Mathematics and the People

Organizers

Abram Kaplan

Harvard Society of Fellows

Jemma Lorenat

Pitzer College

Chair

Jemma Lorenat

Pitzer College

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Session Abstract

The aim of this session is to discover what historians of modern mathematics can contribute to scholarly literature about popular science. Recent attempts to theorize popular science in Europe have invoked the concept of science as communicative action in order to overcome top-down, diffusion-based models of popularization. Mathematics is a fruitful site to investigate how basic institutions of the Enlightenment-universal reason, the "people," expertise, education, and democracy-were shaped through efforts to create popular science. Focusing on popular mathematics brings such basic features of Enlightenment into question as historical categories that actors themselves raise in the context of efforts to create or identify popular mathematics. On the one hand, an Enlightenment tradition of equating mathematical thought with reason has meant that debates over which mathematics counts as popular ipso facto implicate claims about the nature and distribution of reason. On the other, the increasing specialization of technical-especially mathematical-science through the long nineteenth century saw both top-down and bottom-up efforts to address science to the people, efforts that required establishing both who the people were and what science was. From the establishment of national scientific institutions to educational practices to voting algorithms to forums for published debate, we hope to show that the history of popular mathematics is a central part of the history of the Enlightenment and its aftermath.

Presenter 1

It Is Good That Math Is Hard: Math and Social Theory from Descartes to Rousseau

Abram Kaplan

Harvard Society of Fellows

Abstract

Algorithms are hard to understand. As social critics today recognize, the difficulty of algorithms poses a political problem for a society increasingly dependent on their use. But this problem is not new. On the contrary, math has been hard to understand for a very long time. In this talk I offer a longue-ish durée history of the social ramifications of the difficulty of mathematical thinking, a history stretching from René Descartes' Rules for the Direction of the Mind to theories of voting offered by Jean-Jacques Rousseau and Nicolas de Condorcet. Efforts to mathematize private reason and public discourse went hand-in-hand with efforts to extend mathematical literacy. But these efforts did not always seek to produce more mathematicians. On the contrary, the French-writing philosophers and social theorists who engineered the Enlightenment embraced and exploited the difficulty of mathematics. I argue that the persistent difficulty of mathematics helped maintain a hierarchy between experts and non-experts at the very center of the popular Enlightenment. Eighteenth-century efforts to mathematize voting sought to extend this model into political life by identifying a political expertise whose purpose was to assess popular opinion. The experienced difficulty of technical subjects-whether experienced in salons or educational institutions-is a pillar that supports technical governance.

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Presenter 2

Recreational Mathematics and Mathematics by the People: Folding as a Case Study

Michael Friedman

Humboldt University of Berlin

Abstract

Recreational mathematics was and is a literary genre that, starting already in the 17th century, encouraged readers to practice activities other than just reading: inviting them to practice the sciences in a playful form. Based sometimes on what might seem illusions and tricks, this genre challenged readers to understand what happened in such activities. In that sense, recreational mathematics was conceived during these centuries as a way not only to pique one's curiosity, but also to communicate mathematical knowledge to the literate classes of the population. While Luca Pacioli's 1508 book De viribus quantitatis (On The Power Of Numbers) can be considered one of the first books in this genre, the editorial genre clearly gained in importance during the 17th century, after the publication of Jean Leurechon's Récréation mathématique in 1624. This talk focuses on the last third of the 19th century and the beginning of the 20th century, to see how recreational mathematics functioned in times of growing specialization and abstraction in mathematics. Taking exercises of paper folding as a case study and looking at different mathematicians from France, Germany and the USA, one obtains a strange image of how recreational mathematics functioned. On the one hand this genre, while indeed disseminating mathematics to the general public, presented a simplified version of mathematics. Exercises of paper folding show here that the dissemination also happened outside the mathematical "elite", e.g., in Kindergartens and the school system. On the other hand, recreational mathematics was sometimes not considered as epistemological. The denial of epistemological status to recreational mathematics demands that we take a closer look at what kind of knowledge was actually disseminated with those activities.

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Presenter 3

An Okapi Hypothesis: Survival of the Fittest Geometry in the Monist

Jemma Lorenat

Pitzer College

Abstract

The Monist began publication in 1890 as a journal "devoted to the philosophy of science'' and regularly included mathematical contributions. The audience for the mathematics was understood to be "cultured people who have not a technical mathematical training" but nevertheless "have a mathematical penchant." With these constraints, a uniform and inviting style emerged among the varied contributions, described in contrast to the "very repellent form" of elementary textbooks. The mathematical content varied from recreations to the logical foundations, but everyone had something to say about so-called modern geometry. Complementing studies of non-Euclidean geometry in Europe, a focus on The Monist captures the particular nationalism of the United States at a time when its academic hierarchy was still in flux and mathematical research was just beginning to be recognized abroad. On both sides of the argument, persistent metaphors of stillborn children, extinction, races, and species reflect concurrent discussions in the scientific community around heredity. On one side, George Bruce Halsted ceaselessly advocated the "epoch-making" role of Lobachevsky, Bolyai, and their successors as a return to Greek excellence. Defenders of Euclidean geometry feared that the new geometry would destroy the fundamental strengths of mathematics. Despite ad hominem attacks and name-calling, these exchanges document deeper debates around the role of authority (particularly foreign authorities) in shaping the future of geometry. As one contributor inquired "how is the professional expert better fitted to see more lucidly in dealing with the elements of geometry than any other person of good geometric faculty?"

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Presenter 4

Public Statistics: Making and Subverting Democracy

Theodore Porter

University of California Los Angeles

Abstract

Several other of the most consequential uses of public numbers in the nineteenth and twentieth centuries were designed to ease the burden of calculation. State statistics, the outcomes of laborious counting, appear at first to involve little if any mathematics. The census has the character of a democratic ritual, carried out in order to enhance public knowledge and to foster transparency. Mathematical complexity rarely rears its head in the public reception to census counts and other official numbers. The metric system is another example: here again, the complex work involved in making uniform or comparable numbers disappeared from the product. In more recent times, the standardizing impulse has given us iconic economic and social number such as GDPs, numbers that matter in public debate. Quantitative experts sometimes condemn these numbers as misleadingly simple or as omitting much that really matters most. But iconic numbers like hard to displace or even to critique in public discussion, notwithstanding expert critiques. Meanwhile, earnest efforts to teach people to reason more effectively using probability calculations or other quantitative tools have, in almost every era, fallen flat. This paper explores the role of public statistics in making and subverting democracy.

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