## Math: Still Not Everywhere

*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.

I just read the post from a year ago. I see that the guy has a point, and that it’s close to your (Cathy’s) recurring theme about mathematical authority perpetuating power structures. BUT he does his point a tremendous disservice by completely misinterpreting what people usually mean by “math is everywhere.” As I interpret “math is everywhere,” it has a component orthogonal to his point and a component in agreement with his point, so it’s about 60 degrees away, not 180. The orthogonal component of “math is everywhere” is that math can further understanding of almost everything. The component in agreement is that, as a consequence, math can be connected to power and it’s in your interest to understand it.

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Thanks for raising this Jonathan. I like how you distill those two aspects of “math is everywhere.” Of course one must simplify a bit, but I hope both those aspects came through in the above. The “agreement” component explains why Haensch, Frenkel, and others share many political goals and understandings of the current situation, while the “orthogonal” component tells us why there are such different emphases for what to do about it.

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Thanks for the reply. It’s important to emphasize points of agreement.

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Wow, had to commit serious time to getting through this rather weighty discussion. Neither a math professor or theorist but a believer that most people do not understand even enough math to comprehend compound interest and their own mortgages, promotion of math is something I encourage. However, it still amazed me that professors like David Wootton actually have the chutzpah to criticize something he hasn’t actually read. Wow, no wonder we’re in such a sad state.

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Thanks for sticking with it to the end, Carolyn. As you observe, educational problems often hinge on a mix of philosophy problems and attitude problems, and it’s important to distinguish the two.

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A Layman’s Point of View

Not being a mathematician or historian myself, I too struggled to distill the arguments you were making, but I believe I understood your underlying points…and I agree with them.

It seems many people mistakenly translated the idea that “Math is not Everywhere” to mean not everyone can be exposed to and benefit from math. They can, just not to the levels you are promoting.

A better expression might be; Math “will never” be everywhere, where math is defined not as the ability for everyone to learn and use math, but as the ability for some people to learn and use math to influence and shape the world as they see fit. Another simplistic view might be; “Math is everywhere, but it’s just out of your reach.”

Other examples, money, athletics, and politics, to name a few, share the same distinction. Everyone has access to a job, Little League, or City Council, but for many of the same reasons as math, only the very few have the skills, imagination and advantages to unlock and exploit the power these activities hold.

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Thanks, calcroteaus. That’s a helpful way to think about it. At some level, it comes down to (as the saying goes) what your definition of “is” is. What does it mean to say math “is everywhere” when that everywhere is out of reach? It’s about whether we prioritize the *in principle* or the *in practice*, and that’s not a question that can be resolved philosophically in a vacuum. I also agree with your observation that math is similar to many other sources of power, distinction, etc. — so the next question becomes why it is so much more common to believe (and to insist) that math is everywhere, when we wouldn’t *insist* on corresponding claims about other things? I think that’s what makes this topic especially challenging and important.

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Reblogged this on Curtis Miller's Personal Website and commented:

It’s been a while since I shared something on this site (I’ve been studying for the past month for two qualifying exams, and that consumed all my time). Here I share an excellent article that introduced me to the idea of the “mathematical elite” as a real, socially relevant, and powerful (politically and otherwise) group.

While you’re there, check out the work of Cathy O’Neil (mathbabe). I ready

Weapons of Math Destructionin a day during the spring and found it eye-opening and enthralling. I think anyone who works in quantitatively intense subjects should read that book.LikeLike

I’m glad that someone has the chance to think and reflect on this. Thanks to Michael for responding to comments here.

Unfortunately, I cannot read this without feeling that it is fundamentally disconnected from action on the ground in which numerous people in positions of power are arguing for

lessmathematics being taught, in that expecting people in general to learn math constitutes an unfair and immoral barrier between disadvantaged populations and the certification needed for a sustainably paying job (c.f. Andrew Hacker in the NY Times).Last year, CUNY removed basic 8th-grade level algebra as a requirement for college degrees, and two weeks ago the CA state university system did the same. I realize that you’re talking about the most advanced mathematics, but for these discussions to overlook that the bridge to that point is being actively demolitioned is disappointing.

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Thanks for this comment, Dan, which raises a super important point. For me, one of the hardest parts of the exchange with Frenkel, e.g., was that we generally agreed that cutting back on math education is harmful, but we fundamentally disagreed on how to push back against that. The focus above and in the original piece is on advanced mathematics, you’re right, but I think it also tells us something about, e.g., 8th grade education. Haensch made this explicit: for her (and also I think for Frenkel) the best answer to the “where will I use this?” question is that “math is everywhere”. But I worry that this might even be counterproductive, especially if the examples of math being everywhere are (as they often are) ones whose actual utility seems questionable or contrived. It seems to me far more direct to give an honest answer to “where will I use this?” — and an honest answer will necessarily be a limited one, in many ways. Those who care about math education (myself included) need to get better at saying why it *really* matters that CUNY or CalState graduates can do certain kinds of math with confidence, why pre-meds need a certain mastery of calculus, why someone in a community college technical program needs to factor polynomials (if indeed they do), and so on. Maybe the answer requires rethinking parts of the math curriculum, including expanding or contracting in different areas, but I think it would be a lot more effective and honest a rejoinder than what one often sees from the “math is everywhere” or even “math is the language of nature” or “math is the key to reasoning rigorously” perspective.

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Great answer.

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I was not familiar with the truism “mathematics is everywhere,” and the guest post did not explain it. My guess was that “math is everywhere” could be a system for teaching elementary-school mathematics. For example, start with math in the kitchen: measuring cups are math, oven temperatures are math, timer clocks are math, etc. I went back to the article about Prof. Jordan Ellenberg to find the true definition of “mathematics is everywhere” is that math can give deep, unexpected insight to problems almost everywhere. A surprise can make math more entertaining, but these insights are unexpected because they are rare.

Mathematicians claiming to see insights that other people cannot see is elitist, yet Ellenberg demonstrates non-mathematicians can see those insights, too, after an explanation.

Many things have both an elitist side and a non-elitist side. A musician would claim elite skills beyond most people to apply for a position with a professional orchestra. In contrast. a teacher trying to create a high school orchestra would claim that an orchestra of commonplace students would work fine. Similarly, people playing baseball in the park don’t need to be professional athletes.

My job before I retired was creating data science tools for non-mathematicians. I love Michael J. Barany’s statemet in his Scientific American article about “mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty.” Data science is prone to errors, and we needed our customers to check for those errors with their subject matter expertise. Elitist math would have made my job harder. Fortunately, our algorithms were built on the customers’ procedures, so we could try to explain the limits of the math via the limits of their own techniques.

I had more success at home. My wife and I homeschooled our daughters and they grew up believing, to quote, “Math is what Daddy does. It can’t be hard.” Years later, my younger daughter’s psychology professor asked her to tutor some classmates having trouble with the statistics in psychology. She told me that those classmates were surprised how she stripped all the fuss from the math and taught statistics as practical steps for experiments. Once again, elitism would have been the enemy if she let it in the door.

Michael J. Barany has another stunning revelation above: “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.” Math is like fire. Even math appropriate to the problem must be contained in safe bounds by checking its results against reality.

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Thanks for sharing this comment, Erin!

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