mathbabe

This is a guest post by Michael J. Barany, a postdoc in History at Dartmouth.

One year ago, I wrote a post for the Scientific American Guest Blog arguing against the widespread truism that mathematics is everywhere. The post laid out the history of mathematics as a special and exclusive kind of knowledge wielded by privileged elites. I claimed that the idea that math is everywhere not only gets the history wrong, but also misrepresents how mathematics matters most in most people’s lives, and may be a misguided premise on which to build a more inclusive and responsible discipline. If we start by recognizing the bias and exclusion that affect who gets to use advanced mathematics to intervene in the world, we might get better at responding to those biases while empowering the vast majority in the mathematical non-elite to hold the mathematical elite accountable for the great power they are privileged to wield.

While I received a lot of private responses from people who found the post convincing or clarifying, most of the public reaction represented sharp, sometimes visceral opposition to one or more of my claims. I want to revisit some of those responses here in light of a variety of developments from the past year that I think underscore my argument. The initial responses to my Scientific American post show some of the blindspots and hazards that continue to mark public discourse about mathematics, even from those sincerely committed to rectifying the discipline’s historical inequities. This past year has shown, time and again, that math still isn’t everywhere, and that this matters to everyone.

I wrote my original essay primarily for the mathematics educators, popularizers, and researchers who seem to make the bulk of public claims that “math is everywhere,” as well as for their many different audiences interested in mathematics education and policy. Since I framed my argument historically, however, an important secondary audience came from those who study, share, and read the history of math and science. Their responses bear notice, in part for the contrasts they offer compared to responses from those more vested in the mathematical present.

Specialists on more recent history recognized the essay as a faithful popularization of what recent scholarship on modern science convincingly shows: that math and science are thoroughly political, and that claims to universality are often a misleading means of sidestepping those politics. I received more skeptical responses, however, from some scholars of early periods, especially of the Scientific Revolution. Utrecht University’s Viktor Blasjo got the point of the essay while disagreeing with the interpretation I advanced, calling it a “Po-Mo … party line social constructivist narrative.” (This made Blasjo the second person I know of from 2016 to call my interpretations postmodern as a pejorative. At least my approach is transparent!) If that characterization doesn’t mean much to you, take it simply as an indication of the intensity of interpretive disagreements historians can have while more or less agreeing on the relevant facts.

Other responses showed less careful engagement and less acknowledgement of the grounds for disagreement among historians. A short email with the subject “oh dear” from the University of York’s Anniversary Professor of History, David Wootton, informed me that he suspected I had not read his recent book, which “rather disqualifie[d]” me from opining on the Scientific Revolution. He admitted in a followup email that he had not read my essay, but thought a discussion of it he encountered online seemed “to betray a rather woful [sic] ignorance” that his book would have rectified. The online discussion in question was more substantive than Wootton’s curious irruption of condescension and self-promotion, but reflected a similar dismissive attitude. It began with an extended rebuttal to several points from my essay by math history blogger Thony Christie, who opened by admitting that he was “not really interested in the substantial argument of the article” (a claim Wootton also made to me by email) but rather felt obliged to object to what he saw as historical errors “made worse by the fact that the author is a historian of mathematics.” Since my essay pointed to several places where mathematics was used to claim exclusive authority, and to other places where there were important criticisms of mathematics for precisely that reason, Christie questioned exactly how exclusive mathematics actually was and defended the importance of mathematics in spite of those criticisms. You can decide for yourself from the ensuing back-and-forth what to make of Christie’s objections (I think they only tend to reinforce my argument), but for our purposes here the most significant point was that Christie and Wootton felt comfortable ignoring the stakes and implications of the history of elitism and exclusion in mathematics, as though these were independent of how we understand that history and what we make of it today.

For readers with a stake in today’s mathematics, the response was almost the opposite (and I think a lot more interesting): they tended to grant the historical claims about mathematics and focused almost exclusively on the implications. Two of the more thoughtful and generous responses of this sort came from mathematicians Steven Strogatz and Anna Haensch, on Twitter and the American Mathematical Society Blog on Math Blogs, respectively. Both suggested that we should distinguish the questions of where math is and who is able to use it. This inclination to separate math from its users and creators, I think, gets to the heart of the matter, and was one of my primary reasons for writing the original essay. Haensch argued that “Math is everywhere just as much as anything is everywhere,” that is, that you can find math wherever you look. This view, according to Haensch, “is exactly the antidote” to arguments that dismiss the importance of learning mathematics because of questions like “when will I ever use this?”

Here, Haensch helpfully linked two of the most common kinds of responses to the essay that often appeared less constructively in isolation. The first had to do with what we mean when we say math is everywhere, which in large part is a question about what we mean by math itself. Is math a fundamental latent aspect of the natural world? A basic human capacity for understanding things numerically or logically? An infinitely adaptable tool that modern societies have developed to understand and intervene in the world? A system of training and professionalization that equips certain individuals with specific abstract means of solving problems, and a specific kind of authority that comes with them? How important is it to distinguish basic math from advanced math? Numeracy from algorithms? Dynamical systems from category theory? Is something mathematical if it can in principle be described using mathematics, or just if it is in practice engaged through math?

The second kind of response had to do with the stakes of saying math is everywhere. If math should be more open and inclusive, and more people learning and appreciating math is a good thing, then what do such claims about math accomplish toward that goal? Are there other goals we should have for mathematics that such claims also affect? As many responses put it: does saying math is not everywhere devalue the discipline and make it harder to understand, appreciate, and share? By linking together these two kinds of responses–about what math is, and what is at stake in the answer–Haensch underscored the crucial and fundamental fact that these questions are always implicated in each other: the philosophy of mathematics is political, and the politics of mathematics are philosophical.

Those who claim mathematics is everywhere choose to emphasize what mathematics can be in principle. As Haensch and many others noted, I used Jordan Ellenberg (whose work to share mathematics I greatly respect and admire, for the record) as an example of a mathematician who emphasizes all the places math can reach in order to encourage his audience to appreciate the breadth and power of mathematical thinking. This is not, as many interpreted it, a philosophical claim about the nature of mathematics. Rather (and this is why Haensch’s framing matters), it is primarily a political and pedagogical claim that the best way to understand math (whether or not you’re a mathematician) is as something that is potentially everywhere. And that is what those who expressed either of the two just-considered responses in isolation seemed to miss: that “math is not everywhere” is also at root a political and pedagogical claim, premised on the lessons and legacies of the history of mathematics that most such responses set to the side. Instead of focusing on what math can do in principle, history’s lessons are about what math has been in practice, and this shift in perspective can be as important as the historical episodes themselves.

Which brings us to the most fiery response I received on Twitter, from mathematician Ed Frenkel, whose book Love and Math begins by depicting mathematics as hidden all around us in our daily lives. Frenkel later told me that this was his first real experience of an extended dispute on Twitter, and he would probably have approached it differently in retrospect. Out of respect for this sentiment, I am here focusing on the substance of the exchange rather than the sometimes hyperbolic terms in which it played out. Twitterer @abhinav_shresth distilled one relatively sanitized thread, and a search for both of our Twitter handles shows the parts of the back-and-forth that Frenkel did not delete.

The exchange started after Frenkel shared a link to Haensch’s article, calling it “A good riposte” to my “incoherent ramblings,” and I responded with disappointment that, unlike Haensch, Frenkel seemed to dismiss my essay without taking its claims seriously. It turned out that once Frenkel spelled out his beliefs we had a lot of common ground. We agreed that math is currently and historically elitist and that this is a problem, especially given the huge (and seemingly growing) role mathematics has in contemporary society. Frenkel argued that the solution is for everyone to be empowered by learning more math, to have “equal access” (in his words), and the way to encourage that was to show that math is everywhere. As we have already seen, this claim is both political and pedagogical. Frenkel asserted that the best way to understand the power of mathematics in society is to see it as potentially everywhere, and the best way to give people purchase on that power is to show examples (even mundane ones that are more tractable than the complex mathematics through which that power is often exercised) of mathematics hidden all around us.

By placing the emphasis on how mathematics isn’t everywhere, I claimed that history gave us a different lesson. Politically, I think that it is better to focus on the areas where mathematics does have a profound effect on people’s lives, at the expense of the kinds of tractable examples that are often used to popularize math. This requires sacrificing the expectation that such lessons will always be mathematically tractable, since by their nature these kinds of mathematics are difficult and exclusive, often inaccessible to all but the narrow subset of mathematical professionals who specialize in those specific theories and applications. We should instead seek political, ethical, and other related kinds of understanding about these kinds of mathematics, which would allow more people (however much or little mathematics they know) to hold mathematical elites responsible. Pedagogically, I questioned whether stressing the ubiquity of mathematics was the best motivation. If instead we started by emphasizing that math is and has historically been an alienating and exclusive kind of knowledge (indeed, has often been so by design), then those who have felt alienated or excluded from mathematics need not blame themselves for failing to grasp the mathematics that is supposedly all around them, and mathematics educators (as well as theorists) could prioritize inclusive formulations of their subjects.

A number of developments in the last year have driven home the inadequacy of just trying to convince more people to learn more math as a response to elitism and exclusion in the discipline. News stories abound of malign uses of algorithms and other mathematical technologies for encryption, surveillance, and analysis. During my exchange with Frenkel, one interlocutor called attention to the imminent launch of Cathy O’Neil’s Weapons of Math Destruction book. Indeed, among the book’s many strengths is O’Neil’s way of explaining complex mathematical issues in a way that combines mathematical, political, ethical, and other kinds of understanding. O’Neil used these explanations to argue that the public needs to recognize and appreciate the many specific areas where mathematical models and algorithms affected their lives and society, but that this understanding was not enough. Those who wielded difficult mathematical tools also have a responsibility to use them ethically, and to seek the kinds of mathematical and non-mathematical knowledge that will help them do so.

Amidst the martial analogies associated with O’Neil’s title, there was a striking parallel in both Frenkel’s public writing and a number of reviews of O’Neil’s book to debates about a more common kind of weapon. Back in 2013, Frenkel called for “the 21st century version of the Second Amendment” giving everyone the right “to possess mathematical knowledge and tools needed to protect us from arbitrary decisions by the powerful few in the increasingly math-driven world.” Reviewers of Weapons of Math Destruction, meanwhile, seemed to rush to declare the innocence of mathematics while decrying only those who misuse it out of ignorance or malice. That is: math doesn’t kill people, people misusing math do. And, paraphrasing Frenkel: the only way to stop a bad guy with math is a good guy with math. Decades of policy debates have taught us the dangerous fallacy of these claims when applied to guns instead of mathematics. In a provocative Twitter exchange with mathematician Gizem Karaali, I explored whether the same lessons apply to for math, too. We did not come to a clear conclusion, but the discussion emphasized the importance of asking about responsibility, safety-minded training, and contextual understanding for mathematics education and policy. It also underscored that mathematics, like guns and gun control, is an emotional topic with deep-seated cultural valences that policy-makers ignore at their peril.

If this year has made clear the stakes and power of mathematics in our society, for good and bad, the year has also driven home the range of factors beyond just talent and interest that shape who can wield mathematical power. The end of 2016 saw the theatrical release of the film Hidden Figures, based on Margot Lee Shetterly’s book about African-American women computers at NASA. The film and book were the subject of an especially well-attended panel at the 2017 Joint Mathematics Meetings, and drew attention to how racism and sexism have limited access to advanced mathematical education and careers, while also limiting recognition for those who made major contributions despite those barriers.

The problem is not confined to the past. Shortly after the 2017 JMM, a team of mathematicians launched the excellent and timely Inclusion/Exclusion Blog under the auspices of the American Mathematical Society. Since last February, contributors to that blog have chronicled a wide range of barriers to diversity and access for underrepresented groups in mathematics, as well as a wide range of initiatives aimed at rectifying persistent inequities. These initiatives have often been focused on building professional networks, offering recognition and support, and otherwise promoting mathematicians at an individual and institutional level. Few, as far as I can tell, hinged on the premise that math was everywhere (whether in principle or in practice); most started instead with the unequal realities the discipline currently faces. Contributors have asked tough questions about how to respond to the social and structural conditions that keep mathematics unequal. That linked example was by Piper Harron, who has elsewhere (including on this blog) powerfully analyzed the links between social and structural exclusion and our ideas, assumptions, and approaches to mathematics itself.

The August issue of the Notices of the American Mathematical Society featured a pair of articles on recent political developments in the United States and their affect on the international mathematics community. Another article in the same issue announced a Global Math Project whose aim is to “foster a global conversation about joyous mathematics,” a goal very much in line with the “math is everywhere” approach to access and inclusion: get people excited about math, and inclusion will follow. The juxtaposition with discussions of the U.S. Travel Ban strikingly underscored how access to the mathematical elite is as deeply political as ever, with barriers that require attention to mathematics as a specific and place-delimited discipline rather than a limitless fount of potential joy. While global educational projects can certainly do a lot of good, it is telling that the organizers of this particular project seemed to take for granted that the fundamental problem for mathematics across the globe is “a perception issue,” that it is insufficiently appealing.

There is a potent hope embedded in that kind of thinking. Even if math is and always has been elitist and exclusive, the reasoning goes, it is also (and always has been) available to everyone in principle. The theorems of geometry and the sequence of primes don’t care about where you’re from or the color of your skin. Here, the claim that math can be found everywhere goes hand in glove with the claim that it can be found by everyone. By emphasizing the apparent all-encompassing neutrality of mathematics itself, one might hope, we can see that the only real barriers are the ones we make ourselves and we can resolve to move beyond those barriers individually and collectively. That is, to return to the theme raised by Haensch and Strogatz, by separating math from its users we can aspire to make the user-based practice of math more like math’s universal principles. If we start with “math is everywhere” then we can work toward “math is for everyone” so that any individual has potential ownership of the subject.

I think this gets things backwards. It is precisely because math can’t be separated from its practice and its place in society that, in a meaningful sense, the theorems of geometry do care and have always cared who you are. This is all the more true for math that reaches farther and more powerfully into our lives–the secret mathematics of finance, surveillance, literal weapons and figurative ones–all these kinds of math are guarded and inaccessible by means of an indissociable mix of technical and social barriers. History tells us this isn’t a bug; it’s a feature. It is fundamental to math’s place in the world that it is not open to everyone. But, conversely, the mathematical elite is made collectively, and societies do get to shape who has access to math and what we expect of them. If we start with “math isn’t everywhere,” we are better equipped to see math as embedded in larger social structures of our own making, and, I’d suggest, we are better equipped to reshape those structures for the better.

I may have erred by concluding my Scientific American essay with the implication that “there is much work to be done” so that math might “belong to everyone equally.” There is definitely much work to be done, but that work is premised on the unavoidable reality that math cannot belong to everyone equally, that power does not obey utopian principles. Rather, such inequality creates ethical, political, and pedagogical imperatives, and these latter challenges are what demand constant work and attention. The most misleading aspect of the claim that math is everywhere is its timeless formulation, set apart from movement and change, of opportunities for structural reform. “Math isn’t everywhere” risks that same timeless implication. But math and society alike are always changing, always open to new expectations and understandings. Instead of looking to static universal principles, we might find a more productive kind of inspiration from a recognition more rooted in time and place: math still isn’t everywhere.