## Leah Wright Rigueur: The Republican Party has a Race Problem

Today, Harvard Kennedy School of Government Professor of Public Policy, Leah Wright Rigueur continues her PUP blog series on the role of race in the modern Republican party. Her last piece, cross-posted on the Monkey Cage Blog, was on the surge in Ben Carson’s popularity among Republican voters in a race often dominated by Donald Trump. Today she looks at whether—and how—the “party of Lincoln” can win back black voters. Leah’s recent book, The Loneliness of the Black Republican, offers further insight into the seemingly incongruous intersection of civil rights and American conservatism. We’re delighted to feature her next installment. –PUP Blog Editor

The Republican Party has a Race Problem. Actually, that’s an understatement. The modern Republican Party has a race crisis – one of epic proportions. In the 2012 election, 80 percent of all non-white voters (Black, Hispanic, Asians and other minority groups), voted for President Barack Obama. Nowhere was this more apparent then with black voters – only 6 percent supported Republican nominee Mitt Romney. That’s the lowest amount of support a Republican presidential nominee has received since 1964.

The GOP’s 2013 autopsy report said it best noting that unless the party got serious about tackling the race issue, it would “lose future elections.” And in some respects, some within the party have tried to make racial inroads, particularly among black voters. Unfortunately the Republican presidential primaries have made those outreach efforts a distant memory, as candidates appear to be tripping over themselves to say the most racially offensive things possible. From police brutality jokes, to “media manipulation” comments, to “free stuff” gaffes, the hits keep coming.

The racial gaffes of the primary contenders are a reflection of a party that has a fundamental discomfort with discussing race. That the party has a torturous relationship to racial minorities in 2015 is unsurprising, given that GOP’s public attempts to wrestle with race have been near non-existent except in moments of antagonism.

So where did things go wrong? How did the “Party of Lincoln” fall by the wayside and move so far away from its “civil rights” roots?

The answer isn’t an easy one. Most would point to a long history of racial antagonisms, starting with Barry Goldwater receiving the Republican presidential nomination in 1964. But I’d argue that the disintegration of the relationship goes back even further – just look at Herbert Hoover’s “Lily White” movement or Operation Dixie from the 1950s.

The irony here is that as some within the GOP were hell-bent on alienating non-white voters during this period, others within the party went to great lengths to appeal to racial minorities. That those appeals were effective and coincided with strong (but piece-meal) civil rights decisions from the Republican Party, is a testimony to the support figures like Dwight Eisenhower and Richard Nixon received in 1956 and 1960, respectively.

Yet even as they pursued the non-white vote, Republicans continually weighed this pursuit against the possibility of alienating white southern voters. By 1963, for instance, the GOP had started to exclude African Americans from strategy meetings; a year later, the party had all but eliminated funding for minority outreach efforts and almost all of its black consultants. By the time the party nominated Barry Goldwater for president after a brutal convention battle, racial minorities, especially black voters, had already determined that the GOP offered no sanctuary for racial minorities. Goldwater, after all, was the same senator who had voted against the most comprehensive civil rights act the nation had ever seen. To position such a figure as the face of the Republican Party was a slap, erasing any goodwill the party’s previous efforts had generated.

Richard Nixon had predicted such a disaster, back in 1962, telling Ebony magazine that it would be a mistake for the Republican Party to “accept the beliefs” of Goldwater and “write off the Negro vote.” A Goldwater win, he argued, would mean that the GOP “would eventually become the first major all white political party. And that isn’t good. That would be a violation of GOP principles.”

That Richard Nixon – later of “Southern Strategy” infamy – would make that observation is telling. It speaks to a deep cynicism that invaded the Republican Party prior to Goldwater’s ascent, and took off after Goldwater’s presidential defeat. The next decade and a half would be defined by a party that veered wildly between centrism and right-wing conservatism and a party that fought an ugly, fierce fight over relationship between civil rights and conservatism. Are we an “inclusive or exclusive tent?” was common question among Republican thought leaders throughout the 1960s and 1970s; on the question of race, the party simply could not agree.

For every Edward Brooke in the party there was a Strom Thurmond; more important, was the fact that for every racially progressive initiative, there were at least half a dozen discriminatory actions. As Richard Nixon, for example, poured millions into minority enterprise, historically black colleges and universities, and introduced a progressive Family Assistance Plan, he also cut billions from antipoverty programs, opposed an extension of the Voting Rights Act, and nominated two southerners to the Supreme Court with odious civil rights records. Gerald Ford appointed civil rights lawyer William Coleman as Secretary of Transportation and regularly met with black civic and religious organizations but refused to dedicate significant money and time to minority outreach or racial issues. The GOP’s forceful rejection of any attempt to diversify state and local organizations undercut the party’s rhetoric of a “Big Tent” philosophy. Likewise in 1976, the GOP’s black delegates to the national convention denounced the party’s final platform, alienated by Republicans’ unwillingness to attend to matters of race in a sensitive and thoughtful manner.

But it is Ronald Reagan who offers the most complicated example of the Republican Party’s fractious relationship with race. In 1975 he argued that broadening the GOP base through targeted outreach was a rejection of conservative principles; and in 1976, he ran for president using the now infamous “Welfare Queen” trope. But by 1980, he had changed his mind – somewhat. He and his team employed an approach called “Reagan Focused Impact” (RFI) which relayed conservative messages to target constituent groups while appearing race-neutral. According to campaign memos, those groups were “white, suburbanite ticket-splitters.” Here’s where things get complicated: part of this approach meant campaigning in black and Latino spaces – visiting the South Bronx and talking about economic inequality, for instance – all while simultaneously speaking differently to white southern audiences. The same week that Reagan delivered his infamous “States’ Rights” speech in Philadelphia, Mississippi (the site where 3 civil rights workers had been murdered 16 years earlier), he spoke to the National Urban League. As his strategists explicitly note, these outreach efforts were designed to neutralize black leaders’ outrage while generating positive press among white moderate voters.

In the 30 plus years since then, this approach has guided Republican politics, appearing alongside the more explicit racial gaffes. Sometimes, minority audiences endorse it (Ralph Abernathy did endorse Reagan in 1980), but most of the times, minorities reject the party’s approach, viewing it as insincere and hostile.

The million-dollar question of course, is can the Republican Party win back minority voters? Recent scholarship on the matter doesn’t look promising and the racial “foot-in-mouth” syndrome of the presidential candidates isn’t helping. Neither is the Republican Party’s unwillingness to listen to minority voters on issues of concern or eagerness to advance a revisionist history of the GOP’s relationship to civil rights. Returning to the “Party of Lincoln” isn’t impossible, but it means taking a thoughtful, sensitive approach to racial issues, listening to voter concerns, and endorsing policies and initiatives that reflect those concerns. A small step in the right direction has been the GOP’s interest in mass incarceration reform. At a moment in time when race is going to be central to the 2016 campaign, it remains to be seen if Republicans will address racial issues in a complex, nuanced way that rests on inclusion rather than alienation or exclusion.

Leah Wright Rigueur is an Assistant Professor of Public Policy at the Harvard Kennedy School. She is the author of The Loneliness of the Black Republican: Pragmatic Politics and the Pursuit of Power (Princeton University Press, 2015).

## Wobbly table? Applied math can fix that

We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, new this month. Read his popular first post on color in mathematics here.

In The Princeton Companion to Applied Mathematics (page 50) I mention that a four-legged table provides an example of an ill-posed problem. If we take a table having four legs of equal length lying on a flat surface and shorten one leg by an arbitrarily small amount then the weight supported by that leg will jump from one quarter of the total weight to zero.

A table with one leg shorter than the others wobbles, as may one sitting on an uneven floor, and how to cure wobbly tables has been the subject of a number of papers over the years. The tongue-in cheek article

Hanspeter Kraft, The Wobbly Garden Table, Journal of Biological Physics and Chemistry 1, 95-96, 2001

describes how an engineer, a physicist, and a mathematician would go about solving the problem. The engineer would invent an adjustable leg. The physicist would submit a research proposal to tackle the more general problem of “the stability of multiply-legged objects on rough surfaces”. The mathematician would construct an argument based on the intermediate value theorem to show that stability can be restored with a suitable rotation of no more than 90 degrees. This argument has been discussed by several authors, but turning it into a mathematically precise statement with appropriate assumptions on the table and the ground on which it rests is not easy.

The two most recent contributions to this topic that I am aware of are:

A. Martin, On the Stability of Four-Legged Tables, Physics Letters A, 360, 495-500, 2007

Bill Baritompa, Rainer Löwen, Burkard Polster, and Marty Ross, Mathematical Table Turning Revisited, arXiv:math/0511490v1, 17 pp., 2008

In the latter paper it is shown that if the ground on which a rectangular table rests slopes by less than 35.36 degrees and the legs of the table are at least half as long as its diagonals then the rotation trick works.

For more insight into this problem you may like to watch the video below in which Matthias Kreck explains the problem with the aid of some excellent animations.

## Elizabeth Shakman Hurd on the politics of ‘good religion’ and ‘bad religion’ in international relations

In recent years, North American and European nations have sought to legally remake religion in other countries through an unprecedented array of international initiatives. Elizabeth Shakman Hurd, author of Beyond Religious Freedom: The New Global Politics of Religion, recently had this post appear on the blog, Hamilton and Griffin on Rights, in which she explains what lead her to such a highly politicized and contentious area of study.

Why I Wrote this Book
Elizabeth Shakman Hurd

On the cover of Beyond Religious Freedom is a photo of the desert with a sand berm in the distance and, in the foreground, a line of colorful hand-made flowers sticking haphazardly out of the sand. The Moroccan government built the berm in the 1980s during the war against the Polisario to (literally) draw a line in the sand dividing Moroccan-controlled Western Sahara from the free zone controlled by the Sahrawi Arab Democratic Republic. At 1,600 miles long, it is not surprising that the Sahrawis refer to the berm as al-Jidar, the wall. In stark contrast to the desolation and isolation of the wall, in the foreground of the photo is a row of flowers with cheerful decorated notes attached to their stems. This is a project by Sahrawi artist Moulud Yeslem, Por Cada Mina Una Flor, or For Every Mine, a Flower. Yeslem collects and “plants” the flowers in the desert in peaceful protest against the estimated seven million landmines that are scattered throughout the “no-man’s land” bordering the wall. Each flower has a note attached with a message of solidarity for the Sahrawi people.

The sharp visual contrast between the form of politics represented by the wall and Yeslem’s modest popular protest movement sets the stage for the book’s analysis of the contemporary global politics of religion. Echoing the seemingly unbridgeable distance between the berm and the flowers, I wrote Beyond Religious Freedom to draw attention to the gap between the powerful constructs of religious governance —religious freedom, religious outreach, disestablishment, and interfaith dialogue —authorized by states, experts, and others in positions of power, and the lived experiences of the individuals and communities that they aspire to govern, reform and redeem. The book charts the disjuncture, exclusions, and tensions between the large-scale social, legal and religious engineering projects that have come to dominate the global ‘religion agenda,’ and the lived realities and responses of the individuals and communities that are subjected to these utopian—and often dystopian—efforts. Like the wall, which serves to divide and control the Sahrawi population by reducing their mobility, these projects also divide and discriminate, often in the interests of those in power.

To access this complex field of religio-politics, I present an analytical framework distinguishing between religion as construed by those in positions of legal and political power (“official” or “governed religion”); religion as construed by experts who generate policy-relevant knowledge about religion (“expert religion”); and religion as lived and practiced by ordinary people (“lived religion”). Opening up the study of religion and politics challenges the prevailing assumption in elite academic, legal and policy circles that the legalization of freedom of religion, engagement with faith communities, and protections for religious minorities are the keys to emancipating society from persecution and discrimination. Rather, these efforts exacerbate social tensions by transforming religious difference into a matter of law, enacting a divide between the religion of those in power and the religion of those without it. This leads to forms of politics and public order defined by religious difference, favors forms of religion authorized by those in positions of authority, and excludes other ways of being and belonging, both individually and communally. The book considers a series of pressing questions at the intersection of religion, law, and governance from this angle, including the politics of ‘good religion’ and ‘bad religion’ in international relations; the religion jurisprudence of the European Court; the politics of religious freedom and religious ‘minoritization’ in Turkey, with a focus on the Alevi communities; the politics of sectarianism; and the debates over religious freedom and religious outreach programming in US and European foreign policy.

Beyond Religious Freedom challenges the presumption that academic experts, government officials, and foreign policy experts know what religion is, where it is located, who speaks in its name, and how it should be incorporated into foreign relations. Uninformed assumptions about religion have enabled academics, practitioners and pundits to jump without a second thought into the business of quantifying religion’s effects, adapting religion’s insights to international problem-solving efforts, and incorporating religion’s official representatives into international political decision-making and institutions. Governments, international organizations, and much of the academic literature on religion and international relations treat religion as a relatively stable, self-evident category that is understood to motivate a host of actions, good and bad.

Religion is not however an isolable entity and should not be treated as such, whether in an attempt to separate it from law and politics, or to design a political response to it. Efforts to single out and stabilize religion as a platform from which to develop law and public policy inevitably privilege some religions over others, leading to what Lori G. Beaman and Winnifred Sullivan have described as “varieties of religious establishment.”

One way to access this field is to explore the disjuncture between the forms of religion that are produced by expert knowledge and authorized through legal and governmental practice, on one hand, and the forms of religion lived by ordinary people, on the other. While these fields always overlap, intermingle and shape each other, including institutional religion, they cannot be collapsed entirely.

Discriminating analytically between religious freedom and toleration as construed and implemented by those in positions of power and the life ways of ordinary people provides a unique vantage point on the politics of international religious rights and freedoms. It asks us to consider the lived practices of ordinary people who may have complex and ambivalent relationships to the institutions, orthodoxies, and authorities—both political and religious—that claim to speak on their behalf. What is the impact of legalizing religious freedom on those who dissent from “faiths” as defined by “interfaith” leaders, on those who practice multiple traditions, on those whose practices fail to qualify as a ‘religion’ that merits protection? What are the effects of an expert lobby that insists that states and other authorities construct a legal regime around ‘religious freedom’ and a discursive world around that? Do such projects advance or impede efforts to mitigate violence, discrimination and inequality? Advocates of religious freedom presume that the answer is self-evident and affirmative. Together with a number of others, I find it to be much less certain, and the outcome much less utopian.

Elizabeth Shakman Hurd is associate professor of political science at Northwestern University. She is the author of The Politics of Secularism in International Relations (Princeton) and the coeditor of Comparative Secularisms in a Global Age and Politics of Religious Freedom.

Read a PUP exclusive Q&A with the author, here.

## Vote, or else? Jason Brennan on why moral obligations shouldn’t be enforced

Ethicist Jason Brennan is writing a series of posts for the PUP blog offering unique perspectives on ethics, voting, not voting, democracy, public policy and strategy. He is currently Flanagan Family Associate Professor of Strategy, Economics, Ethics, and Public Policy at the McDonough School of Business at Georgetown University, and is writing Against Politics, under contract with Princeton University Press. You can read his first post on “why smart politicians say dumb things” here–PUP Blog Editor

#### Turnout in American elections is low compared to some other advanced democracies. Should we force people to vote?

Brookings Institute analyst William Galston thinks so. In a recently published Op-Ed at Newsweek, Galston offers a host of arguments on behalf of compulsory voting.[1] None of the arguments are very good.

Galston’s right about one thing: Compulsory voting works. It’s clear that compulsory voting does in fact get more people to vote. But everyone agrees that alone isn’t enough to justify compulsory voting. A basic tenet of liberal democracy, or, really, fundamental human decency, is that it’s wrong to force people to do anything without a strong justification for doing so. Thus, proponents of compulsory voting bear a strong burden of proof. They must produce some reason why it’s permissible to force people to vote.

Does Compulsory Voting Lead to Moderation?

Galston argues that moderates are underrepresented. People belonging to ideological extremes are much more likely to vote than people with middle-of-the-road views. He claims that compulsory voting would thus lead to more moderate political outcomes.

He’s right that moderates vote less. Ample empirical work (e.g., see Ilya Somin’s Democracy and Political Ignorance for a review) shows that political moderates participate less than people with more extreme views. But, that same work also shows that this is because political moderates care less about politics, hold their beliefs more weakly, and also are less informed about politics.

But does compulsory voting actually lead to more moderate political outcomes? The available research (e.g., see Sarah Birch’s Full Participation for a review of the empirical literature) does not support this result. Perhaps it’s because the extremes already tend to balance each other out, and what we actually get from Congress or the president are moderate outcomes and compromise positions.

Indeed, it’s not clear compulsory voting does much of anything. It has no significant effect on individual political knowledge, individual political conservation and persuasion, individual propensity to contact politicians, the propensity to work with others to address concerns, participation in campaign activities, the likelihood of being contacted by a party or politician, the quality of representation, electoral integrity, the proportion of female members of parliament, support for small or third parties, support for the left, or support for the far right.[2]

Is Voting an Enforceable Duty?

Galston believes you have a duty to vote. I disagree,[3] but suppose he’s right and you do have a duty to vote. It doesn’t follow from the mere fact that something is a moral obligation that it’s permissible to force people to do it.

On the contrary, many moral duties—aside from duties to avoid violating others’ rights—are unenforceable. You might have moral duties to keep promises, to be nice to strangers, to buy your mom a birthday present, to be faithful to your boyfriend or girlfriend, to give to charity, to improve your moral character, to apologize for your past wrong-doing, to avoid becoming a member of the KKK, and to avoid using racist language. Nevertheless, these moral obligations are unenforceable—it would be wrong for the government to force you to fulfill these duties, even though they are (Galston and I both agree) moral duties.

So what makes voting special? Why is it an enforceable duty, rather than an unenforceable duty?

Galston says that voting is an expression of gratitude, which makes his defense of compulsory voting all the more perplexing. We often owe it to each other to express gratitude. If you buy me a present, I should say thanks. But in general, the duty is express gratitude is unenforceable. If I don’t send you a thank you note, you shouldn’t call the police and ask them to throw me in jail.

The Public Goods Argument: Are Non-Voters Free Riders?

In an earlier New York Times Op-Ed, Galston describes non-voters as free-riders: “Requiring people to vote in national elections once every two years would reinforce the principle of reciprocity at the heart of citizenship.[4] The idea here is that people who don’t vote are like people who don’t pay their taxes. Non-voters benefit from the good government provided for them by voters, but they don’t do their part in helping to provide that good government. That’s unfair. So, just as it’s permissible to force everyone to pay her fair share of taxes, maybe it’s permissible to force everyone to pay for good government by voting.

On the contrary, I think Galston has an overly narrow view of how citizens fulfill their civic obligations.

Imagine Superman were real. Now imagine Superman never votes or participates in politics. Imagine Galston said to Superman, “You’re a jerk. You free ride off of voters’ efforts. You benefit from good government but don’t do your part.” Superman could respond, “Remember all the times I saved the world? That’s how I did my part.”

Let’s take a less extreme case. Suppose there is a medical genius, Phyllis the Physician. Phyllis is such a genius that she produces new medical breakthroughs hourly. If Phyllis cares about serving the common good, helping her fellow citizens, or paying off some “debt to society”, she has little reason to vote. An hour at the voting booth is worth less than an hour at the lab. Now, imagine Galston said to Phyllis, “You’re a jerk. You free ride off of voters’ efforts.” Phyllis could respond, “No, I’ve paid voters’ back by producing my research. I don’t owe them anything more.”

Superman and Phyllis are extreme cases that illustrate a general point. Each of us in our daily lives as workers, artists, managers, parents, truckers, musicians, priests, teachers, or whatnot, does things that make distant others better off. We’re not just taking; we’re giving. We’re already doing things that make it so that the world and our fellow citizens are better off with us than without us.

There’s no obvious reason to assume that non-voters specifically owe a debt to voters, that the only way we citizens can “pay” for good government is to vote, or that the only way to avoid free-riding on voters’ efforts is to vote ourselves.  If we have a debt to society, or a duty to compensate voters for their efforts, we could instead hold that this debt can be paid, and that voters can be compensated, any number of ways. For any given citizen, given what other citizens are doing and are good at doing, there will be an optimal mix of political and non-political ways for her to pay her debt. For some citizens, this will mean heavy political engagement at the expense of other pursuits. For other citizens, it will mean complete disengagement so as to free the citizen to pursue non-political activities. For most citizens, the optimal mix will be some combination of political and non-political engagement.  Though each citizen might contribute in different ways, they can all pay their debts.

The Best Argument for Compulsory Voting

In the end, the best argument for compulsory voting begins by noting that under a voluntary voting regime, the people who choose to vote are unrepresentative of the population at large.

Voters and abstainers are systematically different. The old are more likely to vote than the young. Men are more likely to vote than women. In many countries, ethnic minorities are less likely to vote than ethnic majorities.[5] More highly educated people are more likely to vote than less highly educated people. Married people are more likely to vote than non-married people.[6] Political partisans are more likely to vote than true independents. In short, under voluntary voting, the voting public—the citizens who actually vote—are not fully representative of the voting eligible public. In general, the privileged are proportionately more likely to vote than the underprivileged. The worry, then, is that because the privileged are more likely to vote, government is likely to be unfairly responsive to their interests. Because the underprivileged are less likely to vote, governments are likely to ignore or underrepresent their interests.

As Galston summarizes the argument:

The second argument for mandatory voting is democratic. Ideally, a democracy will take into account the interests and views of all citizens. But if some regularly vote while others don’t, officials are likely to give greater weight to participants. This might not matter much if nonparticipants were evenly distributed through the population. But political scientists have long known that they aren’t. People with lower levels of income and education are less likely to vote, as are young adults and recent first-generation immigrants.[7]

Let’s put the argument in a more rigorous form. Let’s call this the Demographic Argument for Compulsory Voting:

1.     Voters tend to vote for their self-interest.

2.     Politicians tend to give large voting blocs what they ask for.

3.     When voting is voluntary, the poor, minorities, the uneducated, and young people vote less than the rich, whites, the educated, or older people.

4.     If so, then under voluntary voting, government will tend to promote the interest of the rich, of whites, and of the old, over the interests of the poor, of minorities, or of the young.

5.     Under compulsory voting, almost every demographic and socio-economic group votes at equally high rates.

6.     Thus, under compulsory voting, government will promote everyone’s interests.

7.     Therefore, compulsory voting produces more representative government.

8.     If compulsory voting produces more representative government than voluntary voting, then compulsory voting is justified.

9.     Therefore, compulsory voting is justified.

This argument appears powerful and persuasive at first glance. Nevertheless, as I’ll explain in my next post, it’s unsound. It rests on a number of false empirical assumptions.

Note, however, that Galston cannot consistently advance both the Public Goods and the Demographic Argument for Compulsory Voting. The Public Goods Argument treats voters as cooperators. One person’s vote tends to benefit others, while abstention comes at their expense. The Public Goods argument says that non-voters take advantage of voters. But the Demographic Argument treats voters as competitors. One person’s vote tends to harm other voters (by reducing the power of their vote), while abstention helps them (by strengthening the power of their vote).  The Demographic Argument assumes that non-voters advantage voters, while voters take advantage of non-voters.

At most, one of these arguments is sound. If the Public Goods Argument is sound, then when I (a privileged, upper-middle class, married, white, heterosexual, cisgendered male) abstain, most voters should be mad at me. But if the Demographic Argument is sound, then when I abstain, I do women, blacks, Latinos, the poor, the unemployed, and so on, a favor, by making it more likely the government will represent their interests rather than mine. Galston can’t have it both ways.

[1] http://www.newsweek.com/should-voting-be-compulsory-376905

[2] Sarah Birch, Full Participation: 140; Benjamin Highton and Raymond Wolfinger, “The Political Implications of Higher Turnout,” British Journal of Political Science 31 (1) (2001): 179-223, 179.

[3] See Jason Brennan, The Ethics of Voting (Princeton: Princeton University Press, 2011), chapters 1 and 2.

[4] William Galston, “Telling Americans to Vote, or Else,” New York Times, 6 November 2011, SR9.

[5] In the United States, African Americans typically have a lower overall turnout than whites. However, there is some evidence that, once we control for socioeconomic status and other factors that influence voting turnout, African Americans actually vote in higher rates than whites. For instance, African Americans vote less than whites, because they are more likely to be poor, not because they are African American. However, this probably does not matter for the purposes of the Demographic Argument. See Jan E. Leighley and Jonathan Nagler, “Individual and Systematic Influences on Voter Turnout: 1984,” Journal of Politics 54 (1992): 718-40.

[6] For a review of the empirical literature establishing the claims of this paragraph, see Jocelyn Evans, Voters and Voting: An Introduction (Thousand Oaks: Sage, 2004): 152-6.

[7] Galston, “Telling Americans to Vote”: SR9.

## Patterns are math we love to look at

This piece by Frank Farris was originally published on The Conversation.

Why do humans love to look at patterns? I can only guess, but I’ve written a whole book about new mathematical ways to make them. In Creating Symmetry, The Artful Mathematics of Wallpaper Patterns, I include a comprehensive set of recipes for turning photographs into patterns. The official definition of “pattern” is cumbersome; but you can think of a pattern as an image that repeats in some way, perhaps when we rotate, perhaps when we jump one unit along.

Here’s a pattern I made, using the logo of The Conversation, along with some strawberries and a lemon:

Repeating forever left and right.
Frank A Farris, CC BY-ND

Mathematicians call this a frieze pattern because it repeats over and over again left and right. Your mind leads you to believe that this pattern repeats indefinitely in either direction; somehow you know how to continue the pattern beyond the frame. You also can see that the pattern along the bottom of the image is the same as the pattern along the top, only flipped and slid over a bit.

When we can do something to a pattern that leaves it unchanged, we call that a symmetry of the pattern. So sliding this pattern sideways just the right amount – let’s call that translation by one unit – is a symmetry of my pattern. The flip-and-slide motion is called a glide reflection, so we say the above pattern has glide symmetry.

A row of A’s has multiple symmetries.
Frank A Farris, CC BY-ND

You can make frieze patterns from rows of letters, as long as you can imagine that the row continues indefinitely left and right. I’ll indicate that idea by …AAAAA…. This row of letters definitely has what we call translational symmetry, since we can slide along the row, one A at a time, and wind up with the same pattern.

What other symmetries does it have? If you use a different font for your A’s, that could mess up the symmetry, but if the legs of the letter A are the same, as above, then this row has reflection symmetry about a vertical axis drawn through the center of each A.

Now here’s where some interesting mathematics comes in: did you notice the reflection axis between the As? It turns out that every frieze pattern with one vertical mirror axis, and hence an infinite row of them (by the translational symmetry shared by all friezes), must necessarily have an additional set of vertical mirror axes exactly halfway between the others. And the mathematical explanation is not too hard.

Suppose a pattern stays the same when you flip it about a mirror axis. And suppose the same pattern is preserved if you slide it one unit to the right. If doing the first motion leaves the pattern alone and doing the second motion also leaves the pattern alone, then doing first one and then the other leaves the pattern alone.

Flipping and then sliding is the same as one big flip.
Frank A Farris, CC BY-ND

You can act this out with your hand: put your right hand face down on a table with the mirror axis through your middle finger. First flip your hand over (the mirror symmetry), then slide it one unit to the right (the translation). Observe that this is exactly the same motion as flipping your hand about an axis half a unit from the first.

That proves it! No one can create a pattern with translational symmetry and mirrors without also creating those intermediate mirror symmetries. This is the essence of the mathematical concept of group: if a pattern has some symmetries, then it must have all the others that arise from combining those.

The surprising thing is that there are only a few different types of frieze symmetry. When I talk about types, I mean that a row of A’s has the same type as a row of V’s. (Look for those intermediate mirror axes!) Mathematicians say that the two groups of symmetries are isomorphic, meaning of the same form.

It turns out there are exactly seven different frieze groups. Surprised? You can probably figure out what they are, with some help. Let me explain how to name them, according to the International Union of Crystallographers.

The naming symbol uses the template prvh, where the p is just a placeholder, the r denotes rotational symmetry (think of a row of N’s), the v marks vertical qualities and the h is for horizontal. The name for the pattern of A’s is p1m1: no rotation, vertical mirror, no horizontal feature beyond translation. They use 1 as a placeholder when that particular kind of symmetry does not occur in the pattern.

What do I mean by horizontal stuff? My introductory frieze was p11g, because there’s glide symmetry in the horizontal directions and no symmetry in the other slots.

Another frieze pattern, this one based on a photo of a persimmon.
Frank A Farris, CC BY-ND

Write down a bunch of rows of letters and see what types of symmetry you can name. Hint: the persimmon pattern above (or that row of N’s) would be named p211. There can’t be a p1g1 because we insist that our frieze has translational symmetry in the horizontal direction. There can’t be a p1mg because if you have the m in the vertical direction and a g in the horizontal, you’re forced (not by me, but by the nature of reality) to have rotational symmetry, which lands you in p2mg.

A p2mg pattern based on some of the same raw materials as our first frieze pattern.
CC BY-ND

It’s hard to make p2mg patterns with letters, so here’s one made from the same lemon and strawberries. I left out the logo, as the words became too distorted. Look for the horizontal glides, vertical mirrors, and centers of twofold rotational symmetry. (Here’s a funny feature: the smiling strawberry faces turn sad when you see them upside down.)

One consequence of the limitation on wallpaper groups is that honeybees cannot make combs with fivefold symmetry.

In my book, I focus more on wallpaper patterns: those that repeat forever along two different axes. I explain how to use mathematical formulas called complex wave forms to construct wallpaper patterns. I prove that every wallpaper group is isomorphic – a mathematical concept meaning of the same form – to one of only 17 prototype groups. Since pattern types limit the possible structures of crystals and even atoms, all results of this type say something deep about the nature of reality.

Ancient Roman mosaic floor in Carranque, Spain.

Whatever the adaptive reasons for our human love for patterns, we have been making them for a long time. Every decorative tradition includes the same limited set of pattern types, though sometimes there are cultural reasons for breaking symmetry or omitting certain types. Did our visual love for recognizing that “Yes, this is the same as that!” originally have a useful root, perhaps evolving from an advantage in distinguishing edible from poisonous plants, for instance? Or do we just like them? Whyever it is, we still get pleasure from these repetitive patterns tens of thousands of years later.

Frank A Farris, Associate Professor of Mathematics, Santa Clara University. He is the author of Creating Symmetry.

## Nicholas Higham on Mathematics in Color

We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, which is out this month. A slightly longer version of this post on color in mathematics can be found on Higham’s blog, and it has been cross posted at John Cook’s blog, The Endeavour. —PUP Blog Editor

Color is a fascinating subject. Important early contributions to our understanding of it came from physicists and mathematicians such as Newton, Young, Grassmann, Maxwell, and Helmholtz. Today, the science of color measurement and description is well established and we rely on it in our daily lives, from when we view images on a computer screen to when we order paint, wallpaper, or a car, of a specified color.

For practical purposes color space, as perceived by humans, is three-dimensional, because our retinas have three different types of cones, which have peak sensitivities at wavelengths corresponding roughly to red, green, and blue. It’s therefore possible to use linear algebra in three dimensions to analyze various aspects of color.

## Metamerism

A good example of the use of linear algebra is to understand metamerism, which is the phenomenon whereby two objects can appear to have the same color but are actually giving off light having different spectral decompositions. This is something we are usually unaware of, but it is welcome in that color output systems (such as televisions and computer monitors) rely on it.

Mathematically, the response of the cones on the retina to light can be modeled as a matrix-vector product $Af$, where $A$ is a 3-by-$n$ matrix and $f$ is an $n$-vector that contains samples of the spectral distribution of the light hitting the retina. The parameter $n$ is a discretization parameter that is typically about 80 in practice. Metamerism corresponds to the fact that $Af_1 = Af_2$ is possible for different vectors $f_1$ and $f_2$. This equation is equivalent to saying that $Ag = 0$ for a nonzero vector $g =f_1-f_2$, or, in other words, that a matrix with fewer rows than columns has a nontrivial null space.

Metamerism is not always welcome. If you have ever printed your photographs on an inkjet printer you may have observed that a print that looked fine when viewed indoors under tungsten lighting can have a color cast when viewed in daylight.

## LAB Space: Separating Color from Luminosity

In digital imaging the term channel refers to the grayscale image representing the values of the pixels in one of the coordinates, most often R, G, or B (for red, green, and blue) in an RGB image. It is sometimes said that an image has ten channels. The number ten is arrived at by combining coordinates from the representation of an image in three different color spaces. RGB supplies three channels, a space called LAB (pronounced “ell-A-B”) provides another three channels, and the last four channels are from CMYK (cyan, magenta, yellow, black), the color space in which all printing is done.

LAB is a rather esoteric color space that separates luminosity (or lightness, the L coordinate) from color (the A and B coordinates). In recent years photographers have realized that LAB can be very useful for image manipulations, allowing certain things to be done much more easily than in RGB. This usage is an example of a technique used all the time by mathematicians: if we can’t solve a problem in a given form then we transform it into another representation of the problem that we can solve.

As an example of the power of LAB space, consider this image of aeroplanes at Schiphol airport.

Original image.

Suppose that KLM are considering changing their livery from blue to pink. How can the image be edited to illustrate how the new livery would look? “Painting in” the new color over the old using the brush tool in image editing software would be a painstaking task (note the many windows to paint around and the darker blue in the shadow area under the tail). The next image was produced in
just a few seconds.

Image converted to LAB space and A channel flipped.

How was it done? The image was converted from RGB to LAB space (which is a nonlinear transformation) and then the coordinates of the A channel were replaced by their negatives. Why did this work? The A channel represents color on a green–magenta axis (and the B channel on a blue–yellow axis). Apart from the blue fuselage, most pixels have a small A component, so reversing the sign of this component doesn’t make much difference to them. But for the blue, which has a negative A component, this flipping of the A channel adds just enough magenta to make the planes pink.

You may recall from earlier this year the infamous photo of a dress that generated a huge amount of interest on the web because some viewers perceived the dress as being blue and black while others saw it as white and gold. A recent paper What Can We Learn from a Dress with Ambiguous Colors? analyzes both the photo and the original dress using LAB coordinates. One reason for using LAB in this context is its device independence, which contrasts with RGB, for which the coordinates have no universally agreed meaning.

## The Princeton Companion to Applied Mathematics

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester, and editor of The Princeton Companion to Applied Mathematics. His article Color Spaces and Digital Imaging in The Princeton Companion to Applied Mathematics gives an introduction to the mathematics of color and the representation and manipulation of digital images. In particular, it emphasizes the role of linear algebra in modeling color and gives more detail on LAB space.

## Ethicist Jason Brennan on why smart politicians say dumb things

Ethicist Jason Brennan, whose posts on the ethics of voting for our 2012 Election 101 series were enormously popular, will be writing a series of posts for the PUP blog offering unique perspectives on ethics, voting, not voting, democracy, public policy and strategy. He is currently Flanagan Family Associate Professor of Strategy, Economics, Ethics, and Public Policy at the McDonough School of Business at Georgetown University, and is writing Against Politics, under contract with Princeton University Press. We’re excited to have him back, and to kick it off with his first post. –PUP Blog Editor

#### Saying stupid things to would-be voters is a very smart thing to do.

The Onion jokes: Donald Trump is “an eccentric, megalomaniac billionaire still more relatable to average Americans than anyone willing to dedicate life to politics”. Every other day, he says something outrageous or blatantly false, and yet he continues to grow in the polls. He seems to be getting by on empty slogans, with no well thought out policy ideas.  When you see a politician saying something outrageous or blatantly false, you might be tempted to decry the quality of our politicians. If only someone better came along.

But there’s a reason we have the kind of politicians we do, and it’s not because no one better is willing to step up to the plate. Nor is it because great and evil villains (insert the Koch Brothers or George Soros, depending on your political predilection) are keeping our saviors down. Donald Trump may or may not be an eccentric megalomaniac, and he has indeed said many substantively stupid things. But he’s not a stupid man, and saying stupid things to would-be voters is a very smart thing to do.

Politicians are trying to win elections. To win elections, they need to get the most votes. To do that, they need to appeal to as many voters as possible. In an election, what every smart politician is trying to do is behave in ways that he or she hopes will appeal to the typical voter. Politicians are like this because they respond rationally to the incentives democracy creates.   If voters were well-informed, dispassionate policy-wonks, then political campaigns would resemble peer-reviewed economics journals. But few voters or potential voters are like that. As I’ll document at greater length in future blog posts here, most voters are poorly informed, passionate, biased, overconfident, and tribalistic. Most non-voters are not dispassionate truth-seekers; rather, they just don’t care much at all.

Voters are like this because they respond rationally to the incentives democracy creates. The problem is that our individual votes count for very little. Economists and political scientists debate just how to calculate the probability that your vote will make a difference. Still, even on the most optimistic estimate in the literature, your vote (in a presidential election) has a 1 in 10 million chance of making a difference, but only if you live one of handful of swing states and vote Democrat or Republican. Otherwise, your vote has no real chance of mattering. Polls show that citizens more or less realize this.

Voters do not consume much information, nor do they discipline themselves to think rationally about the information they consume, because their votes make little difference. As economists like to say, voters are rationally ignorant. Consider, as an analogy. Suppose a billionaire offers you a million dollars if you can ace the Advance Placement Economics and Political Science exams. You’d probably be willing to learn basic economics and political science for that price. But now suppose the billionaire instead offers you a 1 in 20 million chance of earning that million dollars if you ace the exams. Now it’s not worth your time—it doesn’t pay to learn economics or political science.

Indeed, it’s not clear that voters are even trying to change the outcome of the election when they vote.  One popular theory of voter behavior is that voters vote in order to express themselves. Though the act of voting is private, voters regard voting as a uniquely apt way to demonstrate their commitment to their political team. Voting is like wearing a Metallica T-shirt at a concert or doing the wave at a sports game. Sports fans who paint their faces the team colors do not generally believe they will change the outcome of the game, but instead wish to demonstrate their commitment to their team. Even when watching games alone, sports fans cheer and clap for their teams. Perhaps voting is like this.

When you see politicians saying dumb things, remember that these politicians are not fools. They are responding rationally to the incentives before them. They say dumb things because they expect voters want to hear dumb things. When you see that voters want to hear dumb things, remember that the voters are only foolish because they are responding rationally to the incentives before them. How we vote matters, but for each individual person, how she votes does not. Thus, most individuals vote as if very little is at stake.Trump’s popularity is an indictment of democracy, not a conviction (yet). Democracy may make us dumb, but that doesn’t mean that in the end, democracies always make dumb decisions.

Jason Brennan is Flanagan Family Associate Professor of Strategy, Economics, Ethics, and Public Policy at the McDonough School of Business at Georgetown University. He is the author of Markets without Limits, with Peter Jaworski (2015), Why Not Capitalism? (2014), Compulsory Voting, with Lisa Hill (2014), Libertarianism (2012), The Ethics of Voting (2011), and A Brief Hisotry of Liberty, with David Schmidtz (2010). He is currently writing Against Politics, under contract with Princeton University Press, and Global Justice as Global Freedom, with Bas von der Vossen.

## Robert Wuthnow, author of IN THE BLOOD, on Farmers’ Faith

Robert Wuthnow, Professor of Social Sciences and director of the Center for the Study of Religion at Princeton University, remarked in a recent interview with PUP that he’d spent most of his career writing about religion, and thus his new book, In the Blood: Understanding America’s Farm Families can seem a departure. But farming has more to tell us about religion than meets the eye. Read on as he contemplates the unique relationship between farming and faith.

Farmers’ Faith, by Robert Wuthnow

In Worst Hard Time writer Timothy Egan describes Depression-era farmers believing God was punishing them for shooting rabbits on Sunday.  Others knelt knee-deep in dust praying fervently for rain.  Theirs was a simple faith:  pray hard, live right, and expect God’s blessings in return.

Farmers’ relationship to God has fascinated writers for centuries.  Biblical narratives tell of shepherds and sheep and gleaners and wheat.  The agrarian ideal that interested Enlightenment writers valued farmers’ particular understanding of nature’s God.  Writers today — Wendell Berry, for example — call attention to the spiritual serenity of farms and fields.

Can we learn something important from farmers?  Do their lives, spent so close to the soil and so dependent on nature, generate insights that may have escaped the rest of us?

I grew up on a farm in a community where everyone believed in God.  I’m sure some of them prayed for rain. I imagine many of them talked to God in the quiet of their fields. But times have changed.  The solitary farmer out hoeing the field is a relic.  Farmers now operate expensive GPS-guided tractors while on-board computers monitor the soil.  How has all that changed farmers’ thinking about God?

Writing In the Blood:  Understanding America’s Farm Families gave me an opportunity to explore farmers’ thoughts on a wide range of topics, including religion.  The book draws on lengthy interviews with two hundred farmers.  They varied in age, gender, region, kind of farming, and religious background.  Some farmed only a few acres; others farmed tens of thousands of acres.

Farmers’ faith is still arguably simple.  It varies from person to person, just as it does for other people.  But it converges on a basic point.  Whatever the language used to describe God, God represents an assurance that things will work out.  And working out does not imply that what happens will be what a person wants.  Praying for rain does not increase the chances that it will rain.  It is just a reminder that God, not you, is in charge.  As one farmer explained, “When you get to thinking you’re running the show, that’s when you’ve got a problem.  God’s got a way of saying, I’ll show you who’s running the show.”

Farmers with this view of God said it was born of hard times – and sustained them in those times.  A farm couple struggling to avoid losing everything a second time said they liked being independent but kept being reminded that they had to trust in God.  Another farmer said he had been so depressed from a farm accident that he prayed to die.  It was hard for him to believe that God was on his side, but it helped knowing that God was there no matter what happened.

The logic in these remarks is similar to the view of God that has been identified in other studies.  Even though a person prays to God or works hard in hopes of pleasing God, the idea is not that what a person does actually causes God to respond in a certain way. A farmer may hope that prayer will bring rain, but the exact nature of that hope has less to do with rain than with being aware of God’s existence and thankful for God’s presence.

Perhaps farmers had an advantage in being aware of God’s existence.  Many of them described something ineffable they could only refer to as “the big picture.”  The big picture was an understanding of life from seeing the crops grow and working with animals.  Farmers knew they played a part in nurturing life.  But they realized their role was only a small part of the big picture.

One more thing:  Sometimes you learn as much from what people don’t say as from what they do say.  Many of these farmers lived in conservative communities.  A few were Tea Party Republicans.  They hated the federal government telling them how to farm.  But they didn’t defend their politics with religious arguments.  And they were fed up with politicians who did.

Robert Wuthnow is the Gerhard R. Andlinger ’52 Professor of Social Sciences and director of the Center for the Study of Religion at Princeton University. He is the author of many books, including Rough Country, Small-Town America, Red State Religion, and Remaking the Heartland (all Princeton).

## Medieval Relativisms by John Marenbon

In a commencement speech at Dickinson College yesterday that focused on the virtues of free speech and free inquiry, Ian McEwan referenced the golden age of the pagan philosophers. But from the turn of the fifth century to the beginning of the eighteenth, Christian intellectuals were as fascinated as they were perplexed by the “Problem of Paganism,” or how to reconcile the fact that the great thinkers of antiquity, whose ideas formed the cornerstones of Greek and Roman civilization, were also pagans and, according to Christian teachings, damned. John Marenbon, author of the new book Pagans and Philosophers, has written a post explaining that relativism (the idea that there can be no objective right or wrong), is hardly a post-modern idea, but one that emerged in medieval times as a response to this tension.

Medieval Relativisms
By John Marenbon

Relativism is often thought to be a characteristically modern, or even post-modern, idea. Those who have looked more deeply add that there was an important strand of relativism in ancient philosophy and they point (perhaps wrongly) to Montaigne’s remark, made late in the sixteenth century, that ‘we have no criterion of truth or reason than the example and idea of the opinions and customs of the country where we are’ as signalling a revival of relativist thinking. But the Middle Ages are regarded as a time of uniformity, when a monolithic Christianity dominated the lives and thoughts of everyone, from scholars to peasants – a culture without room for relativism. This stereotype is wrong. Medieval culture was not monolithic, because it was riven by a central tension. As medieval Christian thinkers knew, their civilization was based on the pagan culture of Greece and Rome. Pagan philosophers, such as Plato and Aristotle, were their intellectual guides, and figures from antiquity, such as the sternly upright Cato or Regulus, the general who kept the promise he had given to his enemies even at the cost of his life, were widely cited as moral exemplars. Yet, supposedly, Christian truth had replaced pagan ignorance, and without the guidance and grace provided for Christians alone, it was impossible to live a morally virtuous life. One approach to removing this tension was to argue that the pagans in question were not really pagans at all. Another approach, though, was to develop some variety of limited relativism.

One example of limited relativism is the view proposed by Boethius of Dacia, a Master in the University of Paris in the 1260s. Boethius was an Arts Master: his job was to teach a curriculum based on Aristotle. Boethius was impressed by Aristotelian science and wanted to remain true to it even on those points where it goes against Christian teaching. For example, Christians believe that the universe had a beginning, when God created it, but Aristotle thought that the universe was eternal – every change is preceded by another change, and so on, for ever. In Boethius’s view, the Christian view contradicts the very principles of Aristotelian natural science, and so an Arts Master like himself is required to declare ‘The world has no beginning’. But how can he do so, if he is also a Christian? Boethius solves the problem by relativizing what thinkers say within a particular discipline to the principles of that discipline. When the Arts Master, in the course of teaching natural science, says ‘The world has no beginning’, his sentence means: ‘The world has no beginning according to the principles of natural science’ – a statement which is consistent with declaring that, according to Christian belief the world did have a beginning. Relativizing strategies were also used by theologians such as Henry of Ghent, Duns Scotus and William of Ockham to explain how some pagans can have even heroic virtue and yet be without the sort of virtue which good Christians alone can have.

These and other medieval relativisms were limited, in the sense that one reference frame, that of Christianity, was always acknowledged to be the superior one. But Boethius’s relativism allowed pragmatically a space for people to develop a purely rational scientific world-view in its own terms, and that of the theologians allowed them to praise and respect figures like Cato and Regulus, leaving aside the question of whether or not they are in Hell. Contemporary relativists often advocate an unlimited version of relativism, in which no reference frame is considered superior to another. But there are grave difficulties in making such relativism coherent. The less ambitious medieval approach might be the most sensible one.

John Marenbon is a senior research fellow at Trinity College, University of Cambridge, honorary professor of medieval philosophy at Cambridge, and a fellow of the British Academy. He is the author and editor of many books, including Abelard in Four Dimensions, The Oxford Handbook of Medieval Philosophy, The Cambridge Companion to Boethius, and Medieval Philosophy: An Historical and Philosophical Introduction.

## Math Drives Careers: Author Louis Gross

Louis Gross, distinguished professor in the departments of ecology, evolutionary biology, and mathematics at the University of Tennessee, is the author, along with Erin Bodine and Suzanne Lenhart, of Mathematics for the Life Sciences. For our third installment in the Math Awareness Month series, Gross writes on the role mathematics and rational consideration have played in his career, and in his relationship with his wife, a poet.

Math as a Career-builder and Relationship-broker

My wife is a poet. We approach most any issue with very different perspectives. In an art gallery, she sees a painting from an emotional level, while I focus on the methods the artist used to create the piece. As with any long-term relationship, after many years together we have learned to appreciate the other’s viewpoint and while I would never claim to be a poet, I have helped her on occasion to try out different phrasings of lines to bring out the music. In the reverse situation, the searching questions she asks me about the natural world (do deer really lose their antlers every year – isn’t this horribly wasteful?) force me to consider ways to explain complex scientific ideas in metaphor. As the way I approach science is heavily quantitative, with much of my formal education being in mathematics, this is particularly difficult without resorting to ways of thought that to me are second nature.

The challenges in explaining how quantitative approaches are critical to science, and that science advances in part through better and better ways to apply mathematics to the responses of systems we observe around us, arise throughout education, but are particularly difficult for those without a strong quantitative bent. An example may be helpful. One of the central approaches in science is building and using models – these can be physical ones such as model airplanes, they can be model systems such as an aquarium or they can be phrased in mathematics or computer code. The process of building models and the theories that ultimately arise from collections of models, is painstaking and iterative. Yet each of us build and apply models all the time. Think of the last time you entered a supermarket or a large store with multiple checkout-lines. How did you decide what line to choose? Was it based on how many customers were in each line, how many items they had to purchase, or whether they were paying with a check or credit card? Did you take account of your previous experience with that check-out clerk if you had it, or your experience with using self-checkout at that store? Was the criterion you used some aspect of ease of use, or how quickly you would get through the line? Or was it something else such as how cute the clerk was?

As the check-out line example illustrates, your decision about what is “best” for you depends on many factors, some of which might be quite personal. Yet somehow, store managers need to decide how many clerks are needed at each time and how to allocate their effort between check-out lines and their other possible responsibilities such as stocking shelves. Managers who are better able to meet the needs of customers, so they don’t get disgusted with long lines and decide not to return to that store, while restraining the costs of operation, will likely be rewarded. There is an entire field, heavily mathematical, that has been developed to better manage this situation. The jargon term is “queuing models” after the more typically British term for a waiting line. There is even a formal mathematical way of thinking about “bad luck” in this situation, e.g. choosing a line that results in a much longer time to be checked out than a different line would have.

While knowing that the math exists to help decide on optimal allocation of employee effort in a store will not help you in your decision, the approach of considering options, deciding upon your criteria and taking data (e.g. observations of the length of each line) to guide your decision is one that might serve you well independent of your career. This is one reason why many “self-help” methods involve making lists. Such lists assist you in deciding what factors (in math we call these variables) matter to you, how to weight the importance of each factor (we call these parameters in modeling) and what your objective might be (costs and benefits in an economic sense). This process of rational consideration of alternative options may assist you in many aspects of everyday life, including not just minor decisions of what check-out line to go into, but major ones such as what kind of car or home to purchase, what field to major in and even who to marry! While I can’t claim to have followed a formal mathematical approach in deciding on the latter, I have found it helpful throughout my marriage to use an informal approach to decision making. I encourage you to do so as well.

## Michael Chwe explains common knowledge, and why it matters to Mark Zuckerberg

Michael Chwe, whose book, Rational Ritual: Culture, Coordination, and Common Knowledge has, in his words, “made its way out of the backwaters of course syllabi” to catch the attention of Mark Zuckerberg, had a terrific piece on the Monkey Cage blog of the Washington Post explaining exactly what common knowledge is, and why it’s so important. According to Chwe, common knowledge is generated by large scale social media platforms like Facebook, and this matters because of the many ways it can be leveraged, among them, stopping violence against women, and helping to foster collective political action.

From his piece on the Washington Post:

When Facebook’s Mark Zuckerberg chose my book “Rational Ritual” last week for his “A Year of Books” book club, I was surprised. “Rational Ritual” came out in 2001, and has somehow slowly made its way out of the backwaters of course syllabi into the elevated spheres of technology companies. This is gratifying to me, because even though it is a scholarly book published by a university press, “Rational Ritual” is essentially a popularization.

“Rational Ritual” tries to popularize the concept of “common knowledge” as defined by the philosopher David Lewis and the sociologist Morris Friedell in 1969. A fact or event is common knowledge among a group of people if everyone knows it, everyone knows that everyone knows it, everyone knows that everyone knows that everyone knows it, and so on.

When I was a graduate student in economics in the late 1980s, most people considered common knowledge as an idea of only theoretical interest. People who thought about collective action (and its flip side, political repression) were mostly interested in the problem of free riding, rather than how people communicate. But social change isn’t just about tackling incentives to free ride – it’s also a problem of coordination.

Recently, Michael Chwe, a master of interdisciplinary applications for otherwise “rarified mathematical theories” has been particularly active in exploring how game theory can help curb sexual violence. Check out his piece on the topic on the PBS Newshour blog here. His recent Q&A with Facebook Books is up here.

## 6 Free to Low-Cost Resources to Teach You Calculus in a Fun and Interactive Way

In his new book, Everyday Calculus: Discovering the Hidden Math All around Us, Oscar E. Fernandez shows that calculus can actually be fun and applicable to our daily lives. Whether you’re trying to regulate your sleep schedule or find the best seat in the movie theater, calculus can help, and Fernandez’s accessible prose conveys complex mathematical concepts in terms understandable even to readers with no prior knowledge of calculus. Fernandez has also provided a list below of his favorite affordable resources for teaching yourself calculus, both on- and offline.

Princeton University Press offers several other books to help you master this most notorious of the mathematics. If you’re already good at calculus, but want to be great at it, check out Adrian Banner’s The Calculus Lifesaver: All the Tools You Need to Excel at Calculus, an informal but comprehensive companion to any single-variable calculus textbook. For high school mathletes and aspiring zombie hunters of all ages, there’s also Colin Adams’s Zombies and Calculus, an interactive reading experience set at a small liberal arts college during a zombie apocalypse. Readers learn as they go, using calculus to defeat the walking dead.

Calculus. There, I said it. If your heart skipped a beat, you might be one of the roughly 1 million students–or the parent of one of these brave souls–that will take the class this coming school year. Math is already tough, you might have been told, and calculus is supposed to be the “make or break” math class that may determine whether you have a future in STEM (science, technology, engineering, or mathematics); no pressure huh?

But you’ve got a little under two months to go. That’s plenty of time to brush up on your precalculus, learn a bit of calculus, and walk in on day one well prepared–assuming you know where to start.

That’s where this article comes in. As a math professor myself I use several free to low-cost resources that help my students prepare for calculus. I’ve grouped these resources below into two categories: Learning Calculus and Interacting with Calculus.

# Learning Calculus.

This online site from Paul Dawkins, math professor at Lamar University, is arguably the best (free) online site for learning calculus. In a nutshell, it’s an interactive textbook. There are tons of examples, each followed by a complete solution, and various links that take you to different parts of the course as needed (i.e., instead of saying, for example, “recall in Section 2.1…” the links take you right back to the relevant section). I consider Prof. Dawkins’ site to be just as good, if not better, at teaching calculus than many actual calculus textbooks (and it’s free!). I should also mention that Prof. Dawkins’ site also includes fairly comprehensive precalculus and algebra sections.

2. Khan Academy–short video lectures (free).

A non-profit run by educator Salman Khan, the Khan academy is a popular online site featuring over 6,000 (according to Wikipedia) video mini-lectures–typically lasting about 15 minutes–on everything from art history to mathematics. The link I’ve included here is to the differential calculus set of videos. You can change subjects to integral calculus, or to trigonometry or algebra once you jump onto the site.

One of the earliest institutions to do so, MIT records actual courses and puts up the lecture videos and, in some cases, homeworks, class notes, and exams on its Open Courseware site. The link above is to the math section. There you’ll find several calculus courses, in addition to more advanced math courses. Clicking on the videos may take you to iTunes U, Apple’s online library of video lectures. Once there you can also search for “calculus” and you’ll find other universities that have followed in MIT’s footsteps and put their recorded lectures online.

If you’re looking for something in print, this book is a great resource. The book will teach you calculus, probably have you laughing throughout due to the authors’ good sense of humor, and also includes content not found in other calculus books, like tips for taking calculus exams and interacting with your instructor. You can read the first few pages on the book’s site.

# Interacting with Calculus.

1. Calculus java applets–online interactive demonstrations of calculus topics(free).

There are many sites that include java-based demonstrations that will help you visualize math. Two good ones I’ve come across are David Little’s site and theUniversity of Notre Dame’s site. By dragging a point or function, or changing specific parameters, these applets make important concepts in calculus come alive; they also make it far easier to understand certain things. For example, take this statement: “as the number of sides of a regular polygon inscribed in a circle increases, the area of that polygon better approximates the area of the circle.” Even if you followed that, text is no comparison to this interactive animation.

One technological note: Because these are java applets, some of you will likely run into technology issues (especially if you’re on a Mac). For example, your computer may block these applets because it thinks that they are malicious. Here is a workaround from Java themselves that may help you in these cases.

Self-promotion aside, calculus teachers often sell students (and parents) on the need to study calculus by telling them about how applicable the subject is. The problem is that the vast majority of the applications usually discussed are to things that many of us will likely never experience, like space shuttle launches and the optimization of company profits. The result: math becomes seen as an abstract subject that, although has applications, only become “real” if you become a scientist or engineer.

In  Everyday Calculus I flip this script and start with ordinary experiences, like taking a shower and driving to work, and showcase the hidden calculus behind these everyday events and things. For example, there’s some neat trigonometry that helps explain why we sometimes wake up feeling groggy, and thinking more carefully about how coffee cools reveals derivatives at work. This sort of approach makes it possible to use the book as an experiential learning tool to discover the calculus hidden all around you.

With so many good resources it’s hard to know where to start and how to use them all effectively. Let me suggest one approach that uses the resources above synergistically.

For starters, the link to Paul’s site takes you to the table of contents of his site. The topic ordering there is roughly the same as what you’d find in a calculus textbook. So, you’d probably want to start with his review of functions. From there, the next steps depend on the sort of learning experience you want.

1. If you’re comfortable learning from Paul’s site you can just stay there, using the other resources to complement your learning along the way.

2. If you learn better from lectures, then use Paul’s topics list and jump on the Khan Academy site and/or the MIT and iTunes U sites to find video lectures on the corresponding topics.

3. If you’re more of a print person, then How to Ace Calculus would be a great way to start. That book’s topics ordering is pretty much the same as Paul’s, so there’d be no need to go back and forth.

Whatever method you decided on, I still recommend that you use Paul’s site, the interactive java applets, and Everyday Calculus. These three resources, used together, will allow you to completely interact with the calculus you’ll be learning. From working through examples and checking your answer (on Paul’s site), to interacting directly with functions, derivatives, and integrals (on the java applet sites), to exploring and experiencing the calculus all around you (Everyday Calculus), you’ll gain an appreciation and understanding of calculus that will no doubt put you miles ahead of your classmates come September.