In summer 2021, we structured our Reading Circle around weekly discussions of Trust in Numbers by Theodore Porter, supplemented by additional readings (see Further reading below). We began this Reading Circle to better understand the historical development of quantitative methods and objectivity as a form of authority, and to investigate the ethical question of how this material ought to be incorporated in the curriculum of STEM students.
Reflecting on our work, we see that as mathematicians, we are members of institutions with a real political stake in the supposed objectivity of mathematics. This book has given us new ways to think about how the rigor of Mathematics (with a capital M) lends authority to questions and beliefs that exist outside our platonic world. The desires for quantification, rigor, and objectivity are not apolitical. Ironically, they are not even objective: “it requires institutional or personal credibility even to produce impersonal numbers.” (Porter, p 214). Reading this book has raised a number of questions for us, many of which we are still sitting with. These are big questions, ones which doers-of-mathematics have an ethical obligation to reckon with.
Here are some of our individual (and course-specific) thoughts.
While reading and discussing this book, I was often reminded that math – even “pure” (non-applied) math – has societal implications around trust, objectivity, and prestige. These are aspects that were not discussed in the math courses I have taken, and topics that I have only hinted at in my own teaching. The main thrust of the book – that quantification and objectivity is a form of control used when trust is absent – has made me think about the ways in which it is vital, and especially appropriate at a liberal arts college like Carleton, to include ethical and sociological considerations as a part of a full mathematical education.
Modeling, in Calculus 2
Throughout the book, Porter reiterates the subjectivity of measurement – what gets measured and what doesn’t, how we do that measurement and whether we trust the measurers, and the ways that we interpret those measurements are all choices that must be made by an expert. Porter says, “We find there is…a willingness to leave untouched the most important issues to deal objectively with those that can be quantified” (Porter, p 229). This is a theme that has come into public awareness recently around, for example, the legitimacy of testing; while scores on an exam are easily calculated and compared, is it actually safe to interpret them as a measure of intelligence, future success, or even mastery of content? This is a theme that is directly relevant to modeling, a topic in the Calculus II (Math 120) curriculum. We ask students to use their knowledge of calculus to interpret models of the real world, and use those interpretations as a tool to aid in understanding real-world phenomena. After reading this book, I have a deeper appreciation for the importance of emphasizing reflection and awareness of these choices in my students.
I plan to create a new mini-project for students in my Math 120 course designed to get students to engage with a specific model, and think critically about what is being measured, what we actually want to know, and how these are related. Additionally, students will dig into what assumptions of uniformity underlie the model; by their nature, models are simplifications of complex systems, and so choices have to be made to ignore some inputs.
Students will engage with these decisions – what is being ignored, why, and what does that mean about the validity of the model? Even deeper: Who makes those decisions, and what are the ways that they leave out (purposefully or not) local expert knowledge?
Mathematical Communication Norms in Mathematical Structures
Porter also discusses the ways that quantification and objectivity serve to stabilize, and legitimize, disciplinary authority. He cites medicine and psychology as examples of fields that used “quantitative rigor … [as] an adaptation to public exposure” (Porter, p 210) and lack of trust. Claims of objectivity serve to shore up shaky foundations, not least in mathematics. However, in practice, the ways that experts communicate with each other are often not as formal as the public face of a discipline suggests: “arguments within a community of specialists can be made with a minimum of formality, only a modest concern for rigor and with frequent recourse to shared, often tacit knowledge” (Porter, p 230). This is also a theme in Bill Thurston’s essay “On Proof and Progress in Mathematics.” He discusses the ways that students of mathematics are often deeply confused when a proof written by a well-regarded expert contains a trivial error – this frequently causes a “compilation error” for the student and prevents understanding, while another expert can see that the error is not fundamental and perhaps even see why that error was made (e.g., the expert is referencing a technique used in another setting in which it is correct).
I intend to make this a central theme throughout my Mathematical Structures (Math 236) course in the Winter Term. This class serves as a launching point for the math major – students learn how to write and read mathematical proofs, and then apply those skills throughout every other class they take. I plan to include direct discussion on mathematical norms around writing: What are they and how do I use them? Why is writing central to mathematics as a discipline and as a community? The class also serves as a way to “socialize” students into the mathematical community; this can be beneficial, since it eases communication and grants students legitimacy in the broader mathematical community, but it can also be detrimental. At its most extreme, the result can be dehumanizing and exclusionary: as an example, Porter points out that “those who survive [the long socialization of graduate study and a post-doc in particle physics] are remarkably homogeneous, not only in scientific commitments, but even in terms of personal habits, mannerisms, and dress” (Porter, p 222). We will also discuss this in class: What is the purpose of a communication norm? Where do they originate, and why?
What are the costs of these norms? For example, what constitutes a “rigorous” proof has changed over time, and is very much dependent on the context of the proof, including its author and the intended audience. We will dig into historical examples (including Ampère’s proof that continuous functions are differentiable, Appel-Haken’s computer-aided proof of the four color theorem, and geometric picture proofs) to discuss what makes these topics and motivate the modern conventions.
As a new faculty member teaching two sections of Calculus 2 in the Fall term, my primary focus in this Reading Circle has been around (1) the question of access to mathematics (as complicated by race, class, gender, disability, proximity to the metropole, etc.) and (2) the self-actualization of students taking mathematics classes, where the latter naturally relates to the former. I wanted to come out of this Summer with clear strategies in mind for how students can think about themselves simultaneously as actors in a larger sociopolitical fabric and as mathematicians, especially given the historical role of calculus content as a gatekeeping tool (to Mathematics and to other STEM majors and to institutions of Academia more broadly) and Calculus 2 in particular as a commonly terminal mathematics course. This Reading Circle has encouraged me to include short excerpts, discussion periods, and reflection prompts designed to challenge students with thinking critically about social roles fulfilled by mathematics, in addition to a more typical strategy for me involving content on ethnomathematics. Taken together, I hope to convey a much more holistic understanding of who does mathematics, the interplays between mathematics and political life, and our role as doers-of-math in an often unquantifiable and dynamic world.
Further Thoughts On Mathematics and Social Positionality
A commonly accepted truth regarding mathematics is its alleged transcending of culture and society as an objective truth-seeking endeavor—see “Western mathematics: the secret weapon of cultural imperialism” by Bishop for a critical analysis of this notion and its role in colonialism—along with the ostensible corollary that mathematics, as a field of study, is an apolitical meritocracy that simply cannot perceive race, gender, sexuality, nationality, or other mundane axes of positionality in its practitioners. How can mathematics ever suffer from issues of prejudice when one can simply check the proof?
Moreover, mathematics is understood (both now and historically) to be a “discipline that disciplines,” (Philips, p 5) one which provides unrivaled mental exercise and unique pathways to self-improvement, even for participants whose day-to-day lives are not manifestly related to mathematics. Mathematics is so understood to be essential to civilized personhood that a tenured Physics professor at the University of Buffalo tweeted recently: “If you don’t understand calculus, you are not a fully educated human.” As put in the popular online textbook, Understanding Calculus by Faraz Hussain, among discussions of free will and cultivating the self:
Therefore, the purpose of studying calculus is two-fold. First is to introduce you to the basic concepts of mathematics used to study almost any type of changing phenomena within a controlled setting. Second, studying calculus will develop invaluable scientiﬁc sense and practical engineering problem solving skills in you. You will understand how to think logically to reduce even the most complex systems to a few interacting components. As you study the main concepts, theories and examples in this book, your mind will develop into a powerful systematic instrument. From “Why Study Calculus.”
The claim that mathematics (and a particular kind of mathematics—calculus) is an essential ingredient for self-actualization is well within the realm of opinions held in the Academy. But why? The main phenomenon investigated within Trust in Numbers is the legitimacy derived across various disciplines and political institutions from rigorous and standardized mathematical study, “in response to a world in which local knowledge had become inadequate.” (Porter, p 93) Among other historical examples, Porter analyzes the elite engineering class of the École Polytechnique in the 19th century—together with the profound impact of its graduates on public life in France—and the political stakes of rigorous mathematical study. Of the institution’s role in legitimizing the old elite in a post-revolutionary world, Porter writes: “In a society that had become suspicious of privilege, meritocracy was a safely elitist form of democracy.” (Porter, p 139) Porter—along with de Sousa Santos, Hacking, and countless other authors—also discuss the role of rigor in the ways that research communities relate within themselves and to one another. Mathematics plays a crucial role in creating social positions.