PLEKTIX

On math and magic

2012-04-29T15:54:00.001-07:00

I've been on a kick lately of re-reading my old favorite fantasy novels. I started with some of Lloyd Alexander's Prydain Chronicles, and am now going back through Ursula K. LeGuin's Earthsea Trilogy. I haven't touched this books—or anything in the fantasy genre—since my early teens, and its been interesting to see how differently I relate to them now.

...from another former obsession

One moment in particular struck me. In LeGuin's A Wizard of Earthsea, there's a scene in which a young apprentice-mage sneaks a look at his master's dusty old spellbooks and becomes transfixed by the ancient runes inside. I realized that the visceral feeling evoked by this passage (and others like it throughout the fantasy genre) is exactly what I felt as a college freshman exploring the math section of my undergraduate science library. I would spend hours at a time browsing dusty old math books, the more arcane the better, trying to decipher their internal logic. Yes, I wanted to learn new math, but I was also hooked on the feeling of being lost in these mysterious tomes. Like the mage's spellbooks, these math books contained strange symbols describing deep and powerful truths, which could only be understood through long, deep study.

A sample from a recent article of mine. Doesn't math look cool?

Reflecting back on these moments highlights how my relationship to mathematics has changed. I was initially drawn to math because of its beauty, elegance, mystery, and because it contained a kind of absolute truth. But after teaching for three years and studying differential geometry for one, I found that abstract beauty and truth were no longer enough to sustain my excitement. I wanted to discover and describe important patterns in the world, not just relationships between abstract constructs. Metaphorically speaking, I wanted to work my magic in the world, not just study it for its own sake. This lead me to study study of complex systems and eventually evolutionary dynamics. Mathematics has lost none of its beauty or mystery for me, but my focus now is on its connection to the world rather than its absolute, self-contained truths.

This parallels, in some ways, the differences I've noticed in the way I approach these fantasy novels now. As a hyper-imaginative pre-teen, I wanted to lose myself in these fantasy worlds, to blur the lines in my mind between these worlds and my own. Re-reading them now, I have no desire to escape into these worlds. Rather I look for metaphors and themes connecting these worlds to mine. These books (and the genre as a whole) seem obsessed with the idea of power: discovering one's own power, learning about different sources of power, coming to grips with the dangers and limitations of power, avoiding the temptation to use power for evil. As a researcher, a future professor, and simply an adult actor in this world, I have a certain measure of real-world power now that I lacked as a bookish pre-teen. In these books, I'm finding an opportunity to reflect on how to wield that power, and the responsibility that comes with it.

Perhaps the larger theme is this: I used to think I needed to escape from the world in order to be myself. Now my goal is to connect to the world, as much as possible, while still being deeply, authentically, myself.

Can we find meaning in evolution?

2012-04-07T20:26:00.001-07:00

I'm a mathematician who studies evolution. I'm also a person who thinks about how people can find meaning and purpose in their lives. And so, combining these, I've spent a fair bit of time thinking about what, if anything, evolution can tell us about the meaning and purpose of human life.

My friend Connor Wood recently wrote on this topic. Specifically, he probed the question of why, precisely, many conservative religious traditions find the idea of evolution so objectionable. His argument is encapsulated in this quote:

I strongly suspect that evolutionary theory makes people so uncomfortable, not because it disagrees with Genesis (lots of things contradict Genesis), but because it presents a vision of a natural world whose “values” are fundamentally opposed to those of our religious cultures.

By "values" (in quotes because evolution is an amoral process), Connor is referring to the often violent struggle to survive and reproduce one's genes, which includes such behavior as infanticide in some mammals and birds. While I agree with Connor's basic argument, I think it's not primarily the violence and struggle that offends some religious sensibilities (the Old Testament and many other religious texts are full of violence) but rather the inherent randomness and lack of ultimate purpose in the process.

Even though scientists generally don't intend it as such, evolution fills the role of a creation story. Like other creation stories, it explains where we came from and how we got here. But unlike other creation stories, it gives us few clues as to where we're going or what we're supposed to do. In fact, it tells us that we're the product of random events. If this randomness had gone differently, we might not be here at all. I think the randomness and lack of purpose implied by this story is why many people—including some who believe it as a scientific hypothesis—find the idea of evolution disturbing.

Where did all this come from?? What does it mean??

Interestingly, several thinkers have tried to turn this equation around, claiming that evolution can, in fact, satisfy our deepest psychological/spiritual needs. One of these is Stuart Kauffman, one of the biggest names in complex systems. Kauffman's latest book, Reinventing the Sacred, argues that evolution is such a creative and fundamentally unpredictable process that it can provide us with all the divine-like inspiration we need.

Unfortunately, Kauffman's idea doesn't quite get there for me. It's true that the variety of life is awe-inspiring, with more and more surprises the closer one looks. However, I think that just being awestruck by the beauty and creativity of nature is insufficient: it doesn't satisfy the questions of why we're here or what we should try to do with our lives.

Another approach is to focus on the potential of evolution to produce cooperation, creativity, and complexity. These aspects of evolution are highlighted in Supercooperators, the new book by my boss and mentor Martin Nowak. I think one of the reasons for the past few decades' surge of research into this side of evolution (the "snuggle for existence") is that it changes the story evolution tells about us, allowing us to understand how love, empathy, and compassion are also products of our evolutionary history.

But I don't find this to be of great philosophical comfort either. First, for every example of the evolution of cooperation, there's a complementary example of evolved selfishness and violence. Second, knowing that my feelings of love and empathy exist because they were successful traits in my ancestors doesn't make me feel better about them. In fact, it makes me feel worse. I want to think of these as fundamental to who I am, not some ploy to reproduce my genes. Every time I try to think about all my love and altruism as being a product of evolution, I become sad and want to stop thinking about it. Perhaps I'm just not thinking about it right, but I imagine others may have this difficulty too.

I made a handy (oversimplified) chart to summarize what I think evolution can and can't do for us in terms of filling philosophical/spiritual voids:

In short, my answer is that no, I don't think evolution can provide us with satisfying answers to many of our deepest questions.

Some atheists/materialists argue that the conversation should end here: There is no larger meaning or purpose to life, and any quest for such is a waste of time. But these questions are a real part of who I am, as real as love or anything else I feel. Doubtless, such searchings are products of evolution themselves. Yet to rationalize them away would be to deny a fundamental part of myself. Besides, if life truly has no purpose, then what would my time be better spent doing? Reproducing my genes? Why should I care about that either, if that's also just another artifact of evolution?

My approach is to grapple with these questions head on, knowing that there are no easy answers. Evolution—the most credible scientific theory as to how we got here—doesn't tell us where we're going or what to strive for. And yet it has implanted us with a deep need to plumb these questions. One could, I suppose, see this as a cruel joke that our evolutionary history has played on us. But I think these questions are as real and important as anything else we experience in life, and there is fulfillment and self-knowledge to be found in exploring them, even if we strongly suspect that satisfying answers will never be found.

The Origins of Inequality

2011-11-27T10:54:00.000-08:00

Inequality is a national conversation topic now, thanks largely to the efforts of Occupy Wall Street and the broader Occupy movement. Fundamental questions are being asked, such as "Must inequality necessarily be a part of human society?", "Are we genetically disposed toward hierarchy or egalitarianism?", and "What would a global egalitarian human society look like?"

We can gain a bit of perspective on these questions by looking at the evolutionary history of humans and our primate relatives:

Our two closest animal relatives are chimpanzees and bonobos. Chimpanzee society is characterized by a strict hierarchy of males, with frequent aggressive conflicts between them to maintain or challenge dominance order. In bonobo society, on the other hand, hierarchies are weak, and conflict resolution is peaceful, often involving sex play.
Baboon societies in the wild are also characterized by a strict dominance hierarchy, in which higher-ranking males regularly harass lower-ranking ones and commandeer their food or resting spots. There is one notable exception, however. In 1982, all of the dominant males in a baboon tribe observed by Robert Saplosky were suddenly wiped out by a tuberculosis outbreak, leaving only the lower-ranking males. There followed a marked shift in the culture of the troop: hierarchy remained, but those at the top were much less likely to harass lower-ranking males or steal their food. Moreover, this more relaxed culture was observed in the same tribe two decades later, even after all the males present during the original shift had died or migrated to other tribes (Saplosky and Share 2004).
Modern hunter-gatherer societies, the closest analogue we know of to our distant ancestors, are uniformly characterized by a strong egalitarian ethos, in which resources are shared and those who attempt to hoard them are ostracized (Boehm 2001). On the other hand, the transition to agriculture lead to the advent of unequal social classes, with the lower classes often suffering from malnutrition (Diamond 1987).

From left to right: Chimpanzee, Bonobo, Olive Baboon. Source: Primate Info Net

Taken together, these examples suggest that humans aren't inevitably predisposed to either hierarchy or egalitarianism. Rather, we are capable of either mode of society. These examples also suggest that, like the baboons, we might be able to shift from one mode to the other in the wake of a destabilizing catastrophe.

A mathematician, economist, or theoretical biologist would call this an example of multiple equilibria. The situation might be depicted like this:

That is, there are two stable configurations of society (really, much more than two, but we're simplifying here): hierarchical and egalitarian. Each equilibrium is stabilized by different mechanisms. In hierarchical societies, those at the top have enough power to squelch any attempt at overthrowing the hierarchy. In egalitarian societies, those who attempt to selfishly amass resources or power are ostracized by the rest of the group. Christopher Boehm discovered these mechanisms for egalitarianim in his survey of modern hunter-gatherer societies

I discovered that their egalitarian political arrangements were quite deliberate. They believed devoutly in maintaining political parity among adults. This belief was so strong that males who turned into selfish bullies, or even tried to boss others around for reasons useful to the group, were treated brutally, as moral deviants. (Boehm 2007)

Because of these mechanisms, the two extreme ends of this spectrum are quite stable. Escaping them is very difficult without a demographic catastrophe like the tuberculosis outbreak in baboons, a major technological shift like the development of agriculture, or a "starting over" opportunity like the colonization of a new continent.

The middle regions of this spectrum, however, are less stable. In these regions, some individuals wield a disproportionate share of power, but not enough to completely suppress the interests of the less-powered classes (the 99%). This leads to persistent power struggles between these classes, in which the balance could ultimately be tipped in either direction.

The United States has always been an unequal society, but the checks and balances of democracy have thus far kept it from sliding into despotic hierarchy. The balance of power has fluctuated throughout our history, with periods of robber baron-style capitalism alternating with progressivist movements. I'm worried, however, that we're currently sliding toward self-reinforcing inequality, as the moneyed elite increase their influence over politics, which leads to policies that make them richer, which gives them even greater influence over politics, and so on.

This multiple equilibria model tells us that we may have only a limited window of opportunity to correct this slide. If an equilibrium of extreme inequality is reached, only an enormous catastrophe would be able to undo it.

Sapolsky, R. and Share, L. (2004). A Pacific Culture among Wild Baboons: Its Emergence and Transmission PLoS Biology, 2 (4) DOI: 10.1371/journal.pbio.0020106

Demographic Transitions and the Future of Humanity

2011-08-02T01:43:00.000-07:00

This week, the field of science bloggers are addressing the question "Are we doomed?" It's a good question. There is no shortage of evidence that we are, in fact, doomed. But as an incorrigible optimist, my response is a cautious "maybe not?"

What am I referring to here? In talking to friends concerned about the future of the world, many express a fear that the human population and its economies will continue to grow until they can no longer be sustained by the resources available on the planet. At this point there will be a great "crunch", as billions die and the rest endure a life of scarcity and strife. This fear is not new; it dates back to Thomas Robert Malthus, who wrote in his 1798 "Essay on the Principle of Population" that

The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction, and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world.

So the Maltusian prediction is yes, we are doomed. But according to contemporary demographic forecasters, Malthus was, surprisingly, wrong. The human population is not growing without bound. Rather, the growth rate is slowing, so that the total population level is headed for a peak—and relatively soon! A 2001 study entitled "The End of World Population Growth" put the chances at 85% that the human population will peak before 2100. Here's a graph of their projections, with the most likely outcomes shaded darkest:

The numbers on the right-hand side are probabilities of being less than some number. For example, there is an estimated 14.4% chance that the population will be less than 6 billion in 2100, and an 89.4% chance it will be less than 12 billion. The thick white line is a UN prediction, not part of this study, provided for the sake of comparison.

This turnaround is remarkable, since the human population has been growing exponentially, with few declines, since the beginnings of recorded history. So what's behind this unprecedented reversal?

It turns out that Malthus didn't know everything about human nature. Population scientists have noticed a surprising, yet robust, pattern in human societies, which they call the "demographic transition". This transition occurs in stages, which are linked to economic and social development.

In pre-industrial societies, parents have many children, since the survival of each individual child is uncertain.
As improvements are made to food supply, hygiene, health care, and infrastructure, more children survive. This leads to a period of rapid population growth.
As the society becomes increasingly urbanized, children become less of an asset (for helping with farmwork) and more of an expense (they must be educated in order to participate in the economy). Increasing education also gives women options other than motherhood. Access to and acceptance of contraception increases. As a net result, birth rates fall.
Eventually, birth rates decrease to levels comparable to or even less than the death rate. The population level then stabilizes or even contracts.

This transition can be seen clearly in the birth and death rates of Sweden:

Lines indicate number of births (blue) and deaths (red) per 1000 people per year.

Note that the death rate in Sweden now exceeds the birth rate. This is also true of most European nations, as well as Canada, Russia, Iran, Japan, China, and many other countries. The United States is an exception for now—but the New England states are an exception to the exception! Many more developing countries have declining birth rates, so that their populations are predicted to stabilize within decades.

The idea of the demographic transition, and its robustness across human societies, gives me hope. Not only because it predicts the end of population growth, but also because it suggests a new paradigm of human existence. A paradigm where quality of life is valued over quantity of life. A paradigm where each individual is cared for, educated, and allowed to dictate the course of his or her own life. A paradigm where the population is stabilized not by coercion, disease, wars or famine (as Malthus predicted) but by the free choices of happy and healthy people.

Of course, there are still major obstacles to overcome before we can live sustainably on this planet. Even as our population stabilizes, irreparable damage to our environment continues. Global consensus remains elusive on challenges such as climate change, deforestation, and biodiversity loss. And conflict, inequality, and oppression squander much of our global human potential.

But the existence of the demographic transition suggests a goal for humanity:

We must facilitate and manage the demographic transition across human societies, in an environmentally sustainable way.

If we achieve this goal, we will be on our way toward a healthy and happy, indefinitely sustainable human population. So maybe, just maybe, we are not, in fact, doomed.

#SciDoom

Freedom and the Public Goods

2011-04-25T08:56:00.000-07:00

Last post, I used the example of a protest against peanut allergy-related procedures to explore how the American conception of "rights" may be changing. In particular, I suggested that ideas of common or collective good were being displaced by an increasingly narrow and selfish definition of individual liberty.

A few friends pointed out that I may have unfairly maligned libertarians, anarchists, and others wary of government power. These people aren't necessarily opposed to volunteerism or helping others; they just don't want to be coerced into doing so (or have their money taken for these purposes).

This is a fair point. However, it doesn't make me feel much better about the "leave me alone" political philosophy. I don't think this philosophy will ever be up to solving our common challenges. To illustrate why, I'd like to bring in a concept from game theory.

The Public Goods Game represents situations in which there is a common resource ("public good") that benefits all members of a group. The public good might be a clean kitchen, a functioning electrical grid, or a healthy environment. This good cannot be maintained without contributions from some group members. Contributions can be in the form of doing something (washing dishes, working in the community garden, donating to NPR) or not doing something (not littering in a public park, not overfishing a lake).

The dilemma is this: everyone benefits from the public good, but contributions are voluntary. The public goods game has no built-in incentive to contribute, beyond the desire to make things better for everyone. So "free-riders" can benefit from others' contributions without contributing anything themselves.

Both theory and experiment predict that cooperation cannot be sustained in such a game. A typical experimental result looks like this:


Horizontal axis is time (number of game rounds) SOURCE: Fehr and Gaechter (2000)

Contributions decline over time to virtually nothing. This is not because the participants are inherently selfish. Indeed, at the start of the game, many people are inclined to contribute. However, they realize at some point that others are exploiting their generosity. Not wanting the benefits of their hard work to reward those who don't contribute, people eventually stop contributing altogether. This unfortunate outcome has a name: the Tragedy of the Commons.

Economists and social scientists have identified a few mechanisms that can reverse this tragedy. If the participants know each other, and also interact in settings aside from the game, then concern for one's reputation can motivate people to contribute (Milinski et al. 2002, Rand et al. 2009). This helps explain why co-ops can be successful: everyone knows each other. They can reward or punish others based on their contributions to cooking, cleaning, and other tasks.

But what about global challenges like climate change, environmental conservation, and sustainable use of resources? These involve billions of players across the globe, and there are strong financial incentives to exploit the public good for individual gain. Furthermore, it can be difficult to trace problems like pollution or overfishing to the individuals or companies responsible. How these large-scale challenges be solved?

Research has identified only one answer. If the game is too large and complex for individual interactions to maintain cooperation, the solution is for the participants to invest in institutions (Gureck et al., 2006; Sigmund et al., 2010). These institutions must have the power to investigate the actions of individuals, and reward or punish them based on their contributions. In other words, a social contract is needed, together with institutions to enforce it.

Of course, powerful institutions have inherent potential for corruption and abuse. This is what worries libertarians and anarchists. I share that concern. But the solution, in my view, is to build in democratic checks, so that these institutions are as responsive as possible to the people they serve.

It's hardly a perfect solution. Institutions can become entangled with those they should regulate. Democratic checks can be co-opted.

But to solve the largest problems that face humanity, we can't count on good will and personal responsibility alone.

References:

Ernst Fehr, & Simon Gaechter (2000). Cooperation and Punishment in Public Goods Experiments American Economic Review, 90 (4)

Özgür Gürerk, Bernd Irlenbusch, & Bettina Rockenbach (2006). The Competitive Advantage of Sanctioning Institutions Science, 312 (5770), 108-111 DOI: 10.1126/science.1123633

Milinski, M., Semmann, D., & Krambeck, H. (2002). Reputation helps solve the ‘tragedy of the commons’ Nature, 415 (6870), 424-426 DOI: 10.1038/415424a

Rand, D., Dreber, A., Ellingsen, T., Fudenberg, D., & Nowak, M. (2009). Positive Interactions Promote Public Cooperation Science, 325 (5945), 1272-1275 DOI: 10.1126/science.1177418

Sigmund, K., De Silva, H., Traulsen, A., & Hauert, C. (2010). Social learning promotes institutions for governing the commons Nature, 466 (7308), 861-863 DOI: 10.1038/nature09203

Peanut Allergies and the Future of Democracy

2011-03-26T08:43:00.000-07:00

Parents are picketing a school in Edgewater, Florida because of restrictions the school put in place to protect a child with a peanut allergy (thanks to my sister for alerting me):

To summarize: a number of parents apparently feel that common-sense measures such as regular hand-washing (which the school is legally required to enforce by the Americans with Disabilities Act) are too onerous to bear. They are demanding that instead this child be removed from the school.

What's going on here? First of all, the claim that these procedures are taking away from educational time is ridiculous. A child with even a moderate behavior problem will waste far more instructional time than these hand-washing procedures ever could. But disruptive students aren't targeted for picketing by parents. Then there's the distraction created by the protesters themselves, which I'm sure is seeping into the classroom.

So it's not about educational time. What is it about then? I can't read these parents' minds, but there are disturbing clues in the language that some of them are using. "They're trying to take away all our rights," says one parent, while a sign reads:

To which rights are they referring, exactly? The Right of Sullied Hands? The Rights of the Unwashed Masses?

This picture is so disturbing because the methods and language of democratic civil rights movements are being used to sacrifice the educational rights of one child so that others can be spared a few minor inconveniences. In this way, these protests are part of a larger, unsettling pattern. The past few years (or perhaps decades?) have seen a subtle shift in the way that terms such as "rights", "justice" and "democracy" are invoked in the US. More and more, these terms are being used to defend indvidual, rather than collective interests, and these individual interests are defined in increasingly narrow and selfish ways.

Mark Lilla, in a New York Review of Books article, sums this up brilliantly:

Many Americans, a vocal and varied segment of the public at large, have now convinced themselves that educated elites—politicians, bureaucrats, reporters, but also doctors, scientists, even schoolteachers—are controlling our lives. And they want them to stop. They say they are tired of being told what counts as news or what they should think about global warming; tired of being told what their children should be taught, how much of their paychecks they get to keep, whether to insure themselves, which medicines they can have, where they can build their homes, which guns they can buy, when they have to wear seatbelts and helmets, whether they can talk on the phone while driving, which foods they can eat, how much soda they can drink… the list is long. But it is not a list of political grievances in the conventional sense.

Historically, populist movements use the rhetoric of class solidarity to seize political power so that “the people” can exercise it for their common benefit. American populist rhetoric does something altogether different today. It fires up emotions by appealing to individual opinion, individual autonomy, and individual choice, all in the service of neutralizing, not using, political power. It gives voice to those who feel they are being bullied, but this voice has only one, Garbo-like thing to say: I want to be left alone.

A new strain of populism is metastasizing before our eyes, nourished by the same libertarian impulses that have unsettled American society for half a century now. Anarchistic like the Sixties, selfish like the Eighties, contradicting neither, it is estranged, aimless, and as juvenile as our new century. It appeals to petulant individuals convinced that they can do everything themselves if they are only left alone, and that others are conspiring to keep them from doing just that. This is the one threat that will bring Americans into the streets.

Welcome to the politics of the libertarian mob.

This all makes me wonder about the future of democracy, and human society more generally. While young people in the Middle East are are staging revolts to overthrow decades of repression and corruption, the Tea Party and Republican Party more generally are attacking the very notion of collective action for the common good. (See "Republicans Vote To Repeal Obama-Backed Bill That Would Destroy Asteroid Headed For Earth" for a satirical example.)

Unsettlingly, the one recent US protest movement on behalf of collective action---the showdown over collective bargaining rights in Wisconsin---has apparently failed. The result is a further weakening of unions, themselves one of the few institutional standard-bearers of the idea that we can achieve more together than apart.

The history of humanity, and indeed of life itself, is a story of transitions from the individual to the collective, from lower to higher levels of organization. This can, of course, be taken too far, as it was in the case of Communism. But happens in the other extreme, of radical individualism and resistance to all forms of organization? Are we headed for an evolutionary regression?

I don't know the answers to these questions, and I don't think they are simple. But since this trend appears to have enormous momentum, it's worth thinking about where it might lead.

(Many of the ideas in this post actually come from my partner Anna. Yay collective action!)

Game theory and Obama's mistake

2010-12-05T21:30:00.000-08:00

Like many of my fellow lefties, I'm disillusioned with current US politics. In my view, we have a president who pursued an admirable, ambitious agenda for two years, but failed to win sufficient public support for his initiatives, and wasted too much time searching for nonexistent common ground. Now, with midterm elections lost, our president seems ready to abdicate all decision making power to the Republicans, whose ideas (in my judgment, as well as in Obama's) will actively make our country worse.

The problem, as I see it, can be illuminated with a bit of game theory. Consider a two-party government, and suppose each party has three possible strategies:

Ideological (I): Fight for initiatives that are consistent with the party's core beliefs, regardless of how popular or achievable these initiatives are.
Pragmatic (P): Work toward compromise and incremental accomplishments, in the view that "mixed bag" policies are better than stalemate.
Cynical (C): Prioritize winning elections and humiliating political opponents over helping the country and upholding core beliefs.

Either side can choose any of the three strategies, giving us nine possible outcomes. Of course, when it comes to the needs of the country as a whole, some outcomes are better than others. The following matrix illustrates (in my judgment) how desirable each outcome is for the country's citizens, on a scale of 0 (horrible) to 9 (awesome).

The zeros—the worst possible outcomes—occur when cynics are allowed to set the agenda. A battle of ideologues vs. cynics isn't much better, but at least the ideologues can stop the worst of the cynics' games. Ideologues vs. ideologues is mostly a stalemate, but the ideologies may overlap enough to allow for cooperation on some fronts.

If both parties are pragmatic, the country gets a solid 7. If one is pragmatic and one is ideological, the outcome depends on how successful the ideology is for the country, hence the wide range of possible values (3-9).

However, politicians aren't only concerned with the needs of the country. They also want to maintain and expand their power. We must therefore also consider how the choices of the parties affect their own success or failure.

This depends in part on how the country as a whole is doing. Let's say times are tough right now: unemployment, wars, etc. Then here's (again, in my own judgment) how the various outcomes will affect the party currently in power:

As you can see, none of the options are great for the incumbents, because whatever happens, the public will tend to (rightly or wrongly) blame them for the current problems. The best they can do is govern well. However, in a US-like system with a supermajority needed to pass any legislation, progress depends on the cooperation of both parties. With a pragmatic opposition, the incumbents can accomplish what needs to be done, obtaining 6's. However, a cynical opposition can play the "stick in the mud" strategy and prevent the government from accomplishing anything. This is bad for the incumbents, because nothing will improve and they will still take the blame.

Here's how it looks for the opposition:

As you can see, the cynical strategy is highly effective for the opposition party. They can stop the wheels of government and, assuming things stay bad, they are virtually guaranteed to win the next election. In game theory terms, C is a Nash equilibrium strategy: C is the best choice no matter what the incumbents do. Of course, "best" here means best for the party, not the country.

In my reading of events, the Republican leadership has decided that C is the way to go. This strategy is examplified by Senate leader Mitch McConnell's statement that "the single most important thing we want to achieve is for President Obama to be a one-term president." There are elements within the Republican party who are more on ideological side, but I see nothing but cynicism from Boehner and McConnell.

Let's say I'm right. Then the Democrats are stuck with these options for their own fortunes:

and these payoffs for the country:

Given these options, the worst possible choice for the incumbent party is P. It's bad for the party because it sets them up to be manipulated by the cynics. It's also bad for the country, because it hands the initiative to those who would sacrifice the country's interests for their own gain.

But, unfathomably, P is exactly what Obama is choosing. This is apparent from his statements such as

Can Democrats and Republicans sit down together and come up with a list of solutions to common problems? I think that we will be able to. I’m doing a whole lot of reflecting, and I think there’s going to be some areas where we need to do a better job.

We see it also from his willingness to accept tax cuts on the wealthiest one percent of earners.

The take-away message from our game theory model is this: There are times when it's good to compromise. If the other side is being pragmatic, or even ideological, compromise can be good for both the country and the party. But there's no point to playing P if your opponent is playing C! The correct response to C is I: counter cynicism by fighting for your core beliefs. Even if the cynics foil your policies, you can thwart their bad ideas and invigorate your supporters.

Some democrats (e.g. Bernie Sanders) have grasped the logic of this situation. But unfortunately, our president isn't yet among them.

The Prisoner's Dilemma on Saturday Morning Breakfast Cereal

2010-10-24T19:53:00.000-07:00

Found on Saturday Morning Breakfast Cereal, via The Astronomist, the cleanest illustration I've seen anywhere of the Prisoner's Dilemma:

The comic ends with a (brief) survey of attempts to convince people to act altruistically rather than selfishly. For me, the more interesting question is how to transform the structure of social interactions, so that altruism is the right choice for individuals as well as for the whole group.

Don't wait for superman

2010-10-18T08:25:00.000-07:00

This weekend I saw "Waiting for Superman" a documentary directed by Davis Guggenheim of Inconvenient Truth fame. It's ostensibly about how great teachers are the key to saving our education system. But what struck me, over and over, was its complete lack of understanding of or regard for what teaching actually entails.

There are many, many problems with this movie, and I will not discuss them all. A website has been set up to debunk it, and on the Daily Kos a classroom teacher provides something of a point-counterpoint. (I should add that I do not necessarily endorse everything said on these sites.) I focus my critique on the movie's conception of teaching, because that's the aspect which clashes most directly with my three years' experience as an urban public school teacher.

My first two years were at the now-defunct Austin Community Academy in Chicago. As an incoming math teacher, I had the good fortune of being mentored by math department chair Steve McIlrath, one of the most amazing and inspiring educators I know. On the day I was hired, he told me was "This may be the most difficult job in America. Every teacher who works here is a hero."

I didn't quite believe him then, but after the first month I knew exactly what he meant. The teachers at Austin were not all amazing educators (especially not me). They were not always flawless in their classroom management or sophisticated in their pedagogy. Personally, I was horrible at classroom management and cringed at my own pedagogy. But just the action of coming in every day to face the students---who were facing their own enormous life challenges---and putting in the effort to manage, engage, and educate them was herioc.

I don't have space to describe how incredibly difficult it is just to be a struggling teacher at these schools, let alone a successful one. If you haven't been there, you don't understand. You can, however, educate yourself through memoirs such as In the Deep Heart's Core, Reluctant Disciplinarian, Chasing Hellhounds, or (ironically enough) Guggenheim's first film The First Year.

Waiting for Superman (WfS) at times acknowledges that teaching is difficult, and that teachers are a "national treasure". But it includes zero interviews with current classroom teachers, and promotes an absurd notion of what teaching is actually about. In one telling moment, a cartoon depicts teachers opening up students' brains and pouring "knowledge" in from a carton. This, we are told, is the way education is supposed to work. Except that now all kinds of standards and regulations have been instituted by various bureaucracies. This multitude of regulations confuses the teacher, who then spills her precious "knowledge" onto the floor.

If this is your picture of teaching, then we can't even begin to talk about education reform. It's not an oversimplification, it's just plain wrong. Educating students---getting them to absorb and engage with new ideas---is what makes teaching hard. This is especially difficult in urban districts where it can be difficult to get students to show up to class, let alone sit politely and receive your teachings. There is no magic carton. Even if there were, students are not mere knowledge repositories but active, thinking beings, and they should be taught as such. Sure, I was operating under many layers of regulation, but these were largely irrelevant to me. What mattered in that room were me, my students, and how I was going to teach them.

WfS's main suggestion for improving our schools is to remove tenure protections so that deadbeat teachers can be fired. These deadbeats are definitely out there. One of them occupied the room right next to Steve's. His idea of music education was to let his students listen to the radio, all day, for the entire year. People like him are criminals. It's deplorable that union contracts prevent the firing of such teachers. I absolutely agree that blanket tenure should be abolished, though there should still be mechanisms to protect teachers from the whims of vindictive principals.

But WfS seems to suggest that removing tenure is the magic bullet needed to fix our education system. This assumes that for every deadbeat fired, there is an excellent teacher waiting in the wings to be hired. That's not the case. As Geoffrey Canada acknowledges during the movie, every excellent teacher starts out as a struggling teacher like I was. These struggling teachers must be thoroughly trained and mentored before and during their first year. All teachers must be given manageable class sizes and courseloads, as well as time to collaborate with their colleagues. They must be given excellent textbooks and other classroom resources. They must be well-compensated so that quality talent is attracted. Schools must be better integrated with social services so that students are healthy and in class every day. Teachers' unions have an important role to play in advocating for teachers' rights and quality of life.

All these reforms are necessary so that struggling teachers can become successful rather than leave the profession (as half do within their first five years). But WfS suggests none of these. Instead, it asserts that all we need is to make teachers more accountable. Trust me, I was already trying as hard as I could. More threats hanging over my head would not have improved my teaching.

Worse, the movie promotes the dangerous idea that we can fix public schools without investing in them. It claims we "tried" spending money and it didn't work, so now we should try something else. This is horrible logic. All of the above reforms require money, along with a good plan for using it. I'm terribly afraid that for years to come, conservatives will cite this movie in their crusade against government spending. Meanwhile, our public schools will continue to languish underfunded.

In short, WfS promotes an absurdly simplistic view of teaching, in which teachers are either good or bad. As soon as we fire the bad ones, we will have only good teachers and top-quality education. This ignores the reality for the vast majority of teachers who are trying but struggling. These teachers are performing one of the the most important and difficult jobs in the country. They need to be supported, and their jobs made more manageable, in order for them to succeed.

I could go on about WfS's other flaws: its bizarre use of pop-culture references (School of Rock??), its incomprehensible reverence for No Child Left Behind, its use of schoolchildren as emotionally manipulative props, etc. But I'll end with this thought: in the closing credits, we see the text "The problem is complex. But the solution is simple."

Take it from a complex systems theorist: this is rarely the case in any context, and it's certainly false when it comes to education reform.

Inside the Mathematician's Studio

2010-08-30T15:30:00.000-07:00

One of the aims of this blog is to give the general public a sense of what we applied mathematicians and other non-laboratory scientists actually do with our time. An earlier post addressed the content of what we do: the development and analysis of models. This post, on the other hand, will focus on process. Specifically, my process: how I actually do math. This post is a joint project with my partner Anna, whose beautiful sequence of illustrated text about the nature of the creative process appears on her site drawmedy.

Of course, there are many aspects of what I do. Activities such as reading through the literature, meeting with collaborators, and writing up results, don't require much explanation. I focus here on the parts of my job that makes me feel most like a mathematician: coming up with new ideas and developing them into mathematical arguments.

It starts with a problem. Most often I'm trying to prove some result of the form "In this model, under these conditions, this kind of behavior can arise". Sometimes these questions can be addressed using textbook-style sequences of steps, or even using programs like Mathematica. But such straightforward solutions don't interest me as a mathematician, and I like to leave this kind of work to other people. What really makes me come alive are the questions for which new mathematical approaches must be conceived.

This is an inherently creative process. There is no way of knowing at the outset what the solution may look like, or even whether a solution will be found. All you start with is your toolbox of mathematical techniques, and some hunches about which tools might work if applied correctly.

From this starting point, it's a process of trying approaches, failing, trying other approaches, asking questions, re-framing the problem, working out simple examples, and trying to make connections between different areas of my knowledge. This process plays out in pencil scratchings on my bound notebooks, two pages of which I've reproduced here:

These two (non-consecutive) pages show some of my musings on Prisoner's Dilemma games played on networks. On the first page I'm mainly working through some visual examples. You can also see some of the general questions these examples inspired. ("Maybe this is all about...")

The first half of the second page shows me asking questions (indicated by the Q:) and formulating hypotheses about how different models might be connected. I typically jot down my thoughts in real time as they occur to me, so that it almost feels like journalling. I tend to write in complete sentences, but sometimes a thought will end mid-sentence as something else occurs to me. I'll also go back and write in the margins (e.g. the circled questions at the top right of the second page) if I have an idea that connects to something I wrote earlier.

The second half of the second page shows some calculations as I test one of the hypotheses generated above. Note the circled line with the words "NOT TRUE" to the right. Mistakes and retractions are ubiquitous in my notebooks (as they probably are in the scratchwork of most mathematicians).

My favorite position for such notebook-scribblings is reclining in a couch or comfy chair, as Anna deftly illustrates:

I tend to get antsy when sitting upright for too long. In fact, I'm a big fan of changing scene in general. If I'm stuck in one room with no good ideas, I'm liable to go searching for another room to work in. Perhaps this helps me get a new perspective on what I'm doing, or maybe it just stops frustration from building up.

I should add that many of my best ideas actually come in the shower, or jogging, or in other situations where my brain has the time and space to chart its own course. Other mathematicians I've spoken to share this experience. If you've been focusing on a single problem for long enough, it can seep into your subconscious, which may continue to generate ideas even when you're doing other things. Back in college (when I was a pure mathematician) I even got to the point of solving homework problems in my sleep, though the sleep was not exactly what you'd call "restful".

I'll end with a call to other science bloggers and writers. The Paris Review, since the 1950's, has conducted a series of interviews with world's preeminent writers on their process: how they generate their ideas and shape them into finished pieces of writing. Collectively, these interviews have helped shape public perception of writing as an occupation, and illustrated the variety of methods that writers employ. In this age where science is increasingly misunderstood and distorted in the public eye, I think it would be powerful to have a similar series of documents illustrating the daily processes of scientists. So I'd encourage any science bloggers/writers reading this to consider expressing your own personal "scientific method" to the general public, and pass the word along!

Eusociality and a blow to kin selection

2010-08-25T20:48:00.000-07:00

A new paper hit the internet today. "The Evolution of Eusociality" by Martin Nowak, Corina Tarnita, and E.O. Wilson re-frames an old evolutionary question and strikes a blow in an increasingly heated debate.

Eusociality is when individual organisms act as a collective reproducing unit. The best-known examples are ants and honeybees, but recently discovered examples include certain beetles, shrimp, and mole rats. Typically all reproduction is done by a single queen, and the rest of the colony exists only to support and protect the queen. Eusociality represents the highest degree of social organization found in nature.

The evolutionary origins of eusociality are something of a puzzle. To transition to eusociality, individuals must give up their own reproductive potential to support that of the queen. This is the ultimate sacrifice, as far as evolution is concerned. If evolution favors those who produce the most offspring, how can it select for actually giving up the chance to reproduce?

The classical answer to this question is kin selection: the idea that cooperative acts can occur between close relatives. Dawkins explained this using the concept of "selfish genes" that promote cooperation with others who have the same gene. One proponent, J.B.S. Haldane, famously said he would jump into a river to save two brothers, or eight cousins.

Ants and honeybees, the two oldest-known examples of eusocial animals, have a special genetic structure in which siblings share 3/4 of their genes, as compared to 1/2 in most sexual reproducers. It seemed reasonable that these close genetic relationships made possible such large-scale organization and extreme altruism.

However, as more eusocial species were discovered, including mammals, this association fell apart. There no longer appears to be any significant relationship between eusociality and relatedness of siblings.

Nowak, Tarnita, and Wilson provide a new model which focuses on the competition between reproductive units, which can be individual or collective. But perhaps more importantly, they thoroughly deconstruct the mathematics underlying kin selection theory.

The big debate in evolutionary theory right now is between those who believe all cooperation can be explained by kin selection (in its more mathematical guise of inclusive fitness theory), and those who believe that the more standard natural selection concept has more explanatory power. This debate has become increasingly heated in recent years.

Backed by rigorous mathematics, the authors argue that

Inclusive fitness theory is not a simplification over the standard approach. It is an alternative accounting method, but one that works only in a very limited domain. Whenever inclusive fitness does work, the results are identical to those of the standard approach. Inclusive fitness theory is an unnecessary detour, which does not provide additional insight or information.

The import of this argument might not be apparent to those not immersed in the field, but this paper could be a turning point in how the evolution of cooperation is understood. Social behavior cannot all be reduced to selfish genes. There are in fact many mechanisms allowing cooperation to evolve. Understanding these mechanisms will continue to be a fascinating question in evolutionary theory.

Is a new mode of evolution emerging?

2010-08-24T15:02:00.000-07:00

Evolutionary theorist Susan Blackmore argues in the New York Times (and elsewhere) that a new form of evolution is emerging, based on the replication of digital information.

This would be the third mode of evolution that we humans are aware of. The first is, obviously, the biological evolution of life. Organisms grow according to DNA blueprints, then produce offspring from copies of these blueprints, perhaps with some variations. Competition between variant copies drives the evolution of life as we know it.

The second mode of evolution is cultural. Ideas spread from person to person, and through this process, whole cultures evolve. Richard Dawkins coined the term "meme" for the units of cultural evolution (i.e. the ideas that "replicate" themselves in people's minds), analagously to genes in biological evolution. Blackmore is a strong proponent of the meme concept, but there is much debate over the utility of this idea in explaining cultural evolution. In any case, it is clear that there are major differences between how biological and cultural evolution work. Understanding and quantifying these differences is a major project for evolutionary theory, and I hope some day to contribute to this effort.

Blackmore calls her proposed third mode of evolution "technological", but "digital" might be a more precise term. Every day, millions of files (encoded in binary) are copied from one location to another. Some files are even programmed to copy themselves. But copying isn't always perfect, and sometimes copies differ slightly from the originals. If these variant copies compete for the ability to reproduce, might we witness a whole new form of evolution in which the "organisms" (which Blackmore calls "temes") are purely digital?

One reason this idea is compelling to me is it follows a pattern of symbolic representations driving changes in the evolutionary process. Biological evolution took off with the advent of DNA/RNA encoding, in which the characteristics of an organism were recorded in an easy-to-copy format. Written language isn't necessary for cultural evolution, but it sure helps. It is much easier to copy the blueprints for, say, a motorcycle, and build new motorcycles from the copied blueprints, than it is to build a new motorcycle by observing an existing one. Symbolic languages facilitate the copying process which is essential for evolution.

Binary is one of the most powerful symbolic languages ever, with the potential to encode almost anything. Binary is also extremely easy (for computers) to copy. It is therefore quite appealing to think that the copying of binary files could form the basis of a new evolutionary process. The artificial life community has been experimenting with this idea for several decades, and I am far too ignorant to comment on their successes and challenges.

I will say that, so far, I can't see much evidence of Blackmore's teme-based evolution happening outside of simulations. The closest parallel seems to be computer viruses, which can copy themselves from computer to computer and sometimes mutate along the way. But these viruses are all designed by humans, and I don't know of any that have evolved novel functionality on their own. Viral videos and other internet memes also rely on the copying of digital information. But the decision to copy such memes is made by humans, so this falls within the domain of cultural evolution.

Will we, in the future, see pieces of code that replicate themselves across the internet, compete with each other, and evolve toward increasing complexity? And if so, will we be able to harness this process for good? Or will it be a mere nuisance, like weeds or spam-bots? I'm not yet convinced that this will happen, but these are important questions to ask.

Update on Game-Based High School

2010-07-29T07:23:00.000-07:00

I wrote a while back on a high school that uses games as its primary pedagogical tool. NPR's All Things Considered has a new report on the school. Excerpt:

"In math, we're traveling around the world," says sixth-grader Rocco Rose, a student at Quest to Learn and a citizen of Creepytown — an imaginary city where his class learns math and English. The students play travel agents, convert currencies, keep blogs about their travel experiences and budget trips.

Creepytown is structured like a video game that has jumped out of the computer. During their 10-week "missions," students learn to adapt and improvise.

"The second trimester, Creepytown went broke," Salen says. "They had ... an economic crisis. So the kids worked to figure out ... what had gone wrong. And then they proposed the design of a theme park to bring revenue in."

Systems Thinking

Salen says playing with complex dynamic systems gives kids opportunities to learn.

Students "learn how to solve problems, how to communicate, how to use data, how to begin to predict things that might be coming down the line," she says.

They also learn something called systems thinking, which Salen says is one of the cornerstones of 21st century literacy. It helps you understand how the behavior of a derivatives trader in Hong Kong affects housing prices in Florida. When a system becomes sufficiently complex, Salen says, you start to get outcomes that are hard to foresee.

"Suddenly you begin to get what's called emergent behavior, and in emergent behavior, that system, the elements in it, begin to relate to one another in ways that can be unpredictable," she says.

Hell yeah! If we can give the next generation early experience with complex systems and unintended consequences, there may be hope for the future yet.

Big Bang Big Boom

2010-07-23T12:36:00.000-07:00

Evolution-inspired animated street art, and one of the most amazing works of art I've seen in any medium:

Quantum Reality and the Measurement Paradox

2010-06-02T20:03:00.000-07:00

I may be primarily an evolutionary theorist nowadays, but I have many interests, and this summer is proving to be a good time to explore some areas not directly connected to my need to publish. Lately I've been doing some reading on quantum mechanics, and what it tells us about reality.

QM is astonishing in both its mathematical elegance and its fundamental counter-intuitiveness. Unfortunately, I think many (including mathematicians) are discouraged from learning about quantum because it is typically presented assuming a deep knowledge of classical mechanics. But in my view, QM isn't just a theory about physics. It's a theory about reality and truth, and many of its implications can be understood with no knowledge of physics at all.

The essential feature of quantum reality, and what makes it different from the way we naturally think, is the superposition principle. It says that if A and B are two possible states of something (a photon, a cat, the whole world...), these states can be added to get another possible state, A+B. For example, if a light switch can exist in ON and OFF positions, there must also be a possible state ON+OFF. Subtraction works too: the state ON-OFF must is a valid state as well. To my mathematician friends: we are moving from the set of possibilities {ON, OFF} to the two-dimensional vector space generated by the basis vectors ON and OFF.

It's important to delineate what is not happening here. ON+OFF does not mean that the switch is stuck somewhere between on and off. It also does not mean that it might be either on or off and we just don't know which. ON+OFF is a fully-determined state which is neither ON nor OFF, but a superposition of the two.

Of course, no one has ever observed a light switch being ON+OFF. Something happens when we observe these superimposed states, such that we can only ever see the "classical" states ON or OFF.

In the standard (a.k.a. Copenhagen) interpretation of quantum mechanics, when a superimposed state is observed, it "collapses" into one of the classically observable states. In the case of ON+OFF, whenever we look at the switch, it collapses into either an ON or and OFF state, with equal probability. But until we look at it, in remains in the state ON+OFF, which has unique properties making it distinct from either the ON or OFF state.

This interpretation poses a host of logical difficulties. What exactly constitutes an "observation", and how would a light switch "know" that it is being observed and should therefore jump into an observable state? Many of the best minds in physics believe that observation has something to do with consciousness, but this raises several obvious questions: How is consciousness is defined? What gives it this unique power to induce jumps in physical states?

I've recently come across a new interpretation, proposed in 1997 by Cerf and Adami. They suggest that superimposed states do not collapse when observed, but rather the observer becomes entangled with the observed, forming a larger superimposed state.

To illustrate this, let's turn to Schrodinger's cat paradox. An atom is prepared in a superposition of two states: one in which the atom will emit a photon and one in which it won't. This atom is placed in a box with a cat and an apparatus which will release poisonous gas if the photon is emitted (the details of the setup are unimportant). According to the Copenhagen interpretation, the system exists in the superimposed state

(EMIT and DEAD_CAT)+(NOT_EMIT and ALIVE_CAT)

until such point as the box is opened by a conscious observer, whereupon the system "collapses" and the cat becomes either just alive or just dead. (This raises some questions of whether cats count as conscious, but such objections only deepen the underlying paradox).

In the Cerf and Adami interpretation, there is no collapse, only entanglement. When we observe the contents of the box, we ourselves become entangled with this system. We become part of the resulting superimposed state:

(EMIT and DEAD_CAT and WE_SEE_DEAD_CAT)
+ (NOT_EMIT and ALIVE_CAT and WE_SEE_ALIVE_CAT)

Of course, we still only see the cat as being either dead or alive, not both. But according to Cerf and Adami, this is only because the state EMIT+NOT_EMIT of the atom is unobservable to us. Of the full superimposed state, we can only see the parts pertaining to the cat and to the observer. Observing only part of the system, it appears to us that the cat is either alive or dead. Anyone else observing the cat would see it to be in the same state that we do, but this is only because the second observer is just as entangled as we are. The cat is still superimposed between alive and dead, and if we could see the whole system, we'd realize that we ourselves are superimposed between seeing it alive and seeing it dead.

From a mathematical point of view, Cerf and Adami's proposal neatly resolves the paradox of observation and state collapse. However, it raises far more troubling questions of its own, which the authors do not begin to explore.

Think of a decision you made today. It's not unreasonable to think that there are quantum processes in our brain whose outcomes affect our decisions (this view is advanced by my friend Bob Doyle). Let's say that there was a certain quantum state in your brain whose collapse into one of two states (in the Copenhagen interpretation) tilted your decision one way or the other.

If this is true, then in Cerf and Adami's interpretation, we actually exist in a superposition of realities: one in which your decision went one way and one in which it went the other. You can only see one of these realities, and everyone you've encountered since has become entangled with you and therefore sees the same reality that you do. But the alternate reality is playing itself out, Sliding Doors-style, superimposed on top of our own.

Furthermore, due to quantum interference, any actions taken in this reality can affect any of the superimposed other realities. And conversely, anything your alternate-reality twin does in his or her reality can affect the reality you and I see.

I tend to believe Cerf and Adami's idea, because millenia of physics research have shown us that the mathematically elegant solution is usually the right one. But this means our universe is weirder than we can possibly imagine.

The Future of Evolutionary Theory?

2010-05-13T14:33:00.000-07:00

Well... it's been quite a month. This April I (a) successfully defended my PhD thesis, and (b) won a Templeton Foundation fellowship to work with Martin Nowak at Harvard for two years. For those who don't know him, Nowak is one of the world's top researchers in abstract evolutionary theory. Working with him will be a tremendous challenge and opportunity.

So how to respond to this challenge? My vision for the next two years is to begin laying out a new mathematical approach to the study of evolution. Allow me to explain.

Currently, the field of evolutionary theory revolves around the study of models. As I discussed a few posts ago, a model takes a real-world situation and reduces it to those features that are considered essential. The model can then be analyzed mathematically, and hopefully the results tell you something useful about the original real-world problem.

Models are powerful tools for understanding the world, but they have a fundamental limitation: they always depend crucially on the particular simplifying assumptions made at the model's inception. A different set of simplifying assumptions might yield completely different conclusions, and it's often unclear which model is more relevant to the natural world.

This problem is ubiquitous in mathematical biology: a paper might devote pages and pages of mathematical analysis to understanding one particular model, but if that model were changed just slightly, all that analysis would suddenly be invalid. The question in my mind is always "What insight do we gain from our mathematics?" All the technical derivation in the world is of limited value unless it can help us reach broader conclusions.

My vision is to shift the focus of evolutionary research from models to theories. A theory, like a model, rests on certain fundamental assumptions, but in the case of a theory these assumptions are so broad as to apply to any system in question. For example, a theory might specify "Individuals interact, reproduce, and die in some manner", whereas a model would have to specify the particular manner in which this occurs. So a single theory can encompass many (even infinitely many) models. It's like the difference between saying "3+4=4+3" versus "x+y=y+x for any real numbers x and y". Moving from models to theories is a leap forward in abstraction, generality, and power.

Shifting to theories also changes the kinds of conclusions you can reach. Models produce predictions: specific outcomes that would occur if reality indeed conformed to the assumptions of the model. Theories produce theorems: general statements that apply to any system of the type in question. A theorem won't tell you exactly what will happen, but it can characterize of the space of possibilities. And that's what I think is needed in evolutionary theory: a general understanding of what can or cannot result from evolution, and how this depends on the certain features of an evolutionary process.

So that's my research agenda in a nutshell. I'm extremely excited to see where this leads, and I'm looking forward to sharing more in the future.

Symbolic Representation is the Key to Major Evolutionary Transitions?

2010-03-28T07:37:00.000-07:00

I'm briefly coming up from the sea of thesis preparation (two weeks until defense!) to share this truly remarkable quote I just read:

Consider the following: in the evolutionary course there
have been a few great junctures, times of major evolutionary
advance. Their hallmark is the emergence of vast, qualitatively
new fields of evolutionary potential, and symbolic representation
tends to underlie such evolutionary eruptions. These "New
Worlds" can arise when some existing biological entity (system)
gains the capacity to represent itself (what it is and/or does) in
some symbolic form. The resulting world of symbols then
becomes a vast and qualitatively new phase space for evolution
to explore and expand. The invention of human language is one
such juncture. It has set Homo sapiens entirely apart from its
(otherwise very close) primate relatives and is bringing forth a
new level of biological organization. The most important of these
junctures, however, was the development of translation, whereby
nucleic acid sequences became symbolically representable in an
amino acid "language," and an ancient "RNA-world" gave way
to one dominated by protein.

-from Carl R. Woese, "On the Evolution of Cells", PNAS, 2002.

Gene-culture Co-evolution

2010-03-02T07:37:00.000-08:00

A while ago, I wrote on the hypothesis that humans have essentially stopped evolving genetically, because of our cultural emphasis on keeping all humans alive, no matter how disadvantaged.

The New York Times reports today on the opposite idea: that human culture may actually intensify the selective pressure on our genes. This idea is known as gene-culture co-evolution, since although our genes and our culture evolve through separate processes (biological reproduction vs. sharing of ideas), these two processes interact and affect each other.

The Times article surveys how culturally evolved changes in diet, lifestyle, and social norms could have influenced the genetic evolution of our digestive systems and brains. But as a discussion starter, I'm interested in more speculative questions: is our evolving culture still shaping our genetic evolution? If so, in which directions are we being pushed?

The Idea of Applied Mathematics

2010-02-25T22:00:00.000-08:00

Mathematicians occupy an odd place in the public imagination, as objects of great curiosity and also great misunderstanding. TV and movies portray us as anything from eccentric to in sane, though sometimes we get to solve crimes. But there is rather little public understanding of what mathematicians actually do with their time.

Even among mathematicians, applied math has an odd reputation. Many pure mathematicians (those who spend their time working on purely abstract problems) regard applied math as mere "computation", as if we were essentially glorified calculators.

But applied mathematics is not about discovering new numbers, nor solving crimes, nor cranking out long calculations (though some of that is involved). At heart, applied math is about creating, refining, and analyzing models.

The "applied" in applied math means that we work on problems that are in some way relevant to the "real world". However, the real world is a complicated place, and virtually any system you might want to investigate has far too many interactions and unknowns to be understood completely. Imagine, for example, trying to understand the physical properties of a gas by first specifying the mass, volume, and exact location of each of billions of molecules, and then trying to predict where each particle will be in the next instant, and then the instant after that. Even if you were somehow able to do all these calculations, your answer would be valid only for that particular gas in that particular configuration, and would give you little insight into the behavior of gases in general.

So when we want to understand a system, we don't attempt to incorporate every potentially relevant detail. Instead, we model it: we focus on what we believe to be the essential features of the problem and throw out everything else. All models are oversimplifications, but if they are well-constructed, that is, if we have picked the right features to keep and the right ones to discard, they can provide valuable insight into the real-world problem we are studying.

All models incorporate a trade-off, which I've (poorly) illustrated here:

We often hear about models on the right end of this spectrum: models of
of large-scale, complex phenomena such as the global climate or economy. These models incorporate many different variables in order to be as accurate as possible in predicting reality. The trade-off is that there is generally less insight to be gained from such models, because cause and effect relationships can be difficult to untangle with so many variables involved.

Mathematicians are more interested in the simple end. Unlike complex models, which can generally only be analyzed through computer simulation, simple models can often be analyzed using a pencil and paper. Though they do not describe reality as accurately as complex models, they illustrate very clearly how and why certain effects lead to certain outcomes. Simple models also have the advantage of generality: the same set of simple features may be present in a wide variety of systems. The more variables and complications you throw in, the more your model becomes tied to the one specific problem you started with.

I've written a lot in this blog about the Prisoners' Dilemma as a model for cooperation. The essence of the model is this: two players each have a choice whether or not to cooperate with the other. If a player decides to cooperate, they pay some cost, and the other player gains some benefit. Of course, cooperation happens in many different forms in human and animal life, and you could study any particular cooperative behavior by tracing its social and/or cognitive basis, as well as its evolutionary origin. But by studying the particularly abstract, simple model that is the Prisoners' Dilemma, you can gain some insight into the phenomenon of cooperation in general: when and why it evolves, and how it is maintained.

The purpose and method of developing and analyzing models is a strangely absent topic from high school and college math and science classes (a welcome exception is a course I'm currently TAing at Boston University that teaches quantitative reasoning to non-science majors). But given the role that models play in our economy as well as in science, and the catastrophic consequences of their failure, I think that communicating an understanding of the modeling process should be a central goal of science education.

Evolutionary Game Theory and Archaeology

2010-01-05T07:59:00.000-08:00

As a mathematical evolutionary theorist, I use abstract methods to investigate how the structure of an evolutionary process determines whether social behaviors like cooperation can be successful. So I was excited to learn over the holidays (from David Carballo, archaeologist and family friend of my partner) that archaeologists are pursuing the same question from an entirely different angle.

As far as I can understand it, there is a new field of research looking at whether evolutionary game theory (EGT) can help explain major societal shifts. One article looks at the sudden appearance of communal architecture projects in Andes mountain societies (in the second and third millenia B.C.E.) that previously had few permanent buildings. These new constructions appear to be built for use by the entire community, and their construction clearly required large-scale cooperation. Using a combination of EGT and historical arguments, the authors posit that the labor for these projects was not coerced. Rather, the chiefs of these societies were able to mobilize cooperation by enforcing norms of fairness and justice. In their words:

Cooperation does not magically emerge. However, when the appropriate conditions are met, cooperation becomes the adaptive choice of people assessing the costs and benefits of participating in specialized versus nonspecialized labor, loss of autonomy, gain in material wealth and nonmaterial benefits, and degree to which the production and redistribution process is “fair.”

While all cooperative systems are vulnerable to "free-riders", who attempt to receive benefits without contributing, the authors argue that the combined mechanisms of punishment and group selection (see this post) were sufficient to overcome this difficulty.

I'm excited to see this field taking off in so many different directions, and I'm looking forward to see what new intersections develop!

Highlights from the Year in Ideas

2009-12-13T17:47:00.000-08:00

The New York Times Year in Review section always has some good ones. Some highlights for me from this year:

Does feeling like a fraud make you act like one? Researchers gave experiment subjects designer-style sunglasses from boxes marked "authentic" or "counterfeit". They then put the subjects in situations with an incentive to be dishonest; far more of the subjects who were told they were wearing counterfeit designer glasses acted in a dishonest manner. Possible conclusion: wearing the "counterfeit" glasses (in reality all the glasses were authentic) made people feel like they were dishonest, and they acted accordingly.

Battle-bots with a moral compass: A roboticist is collaborating with the US army on combat robots (e.g. predator drones) that can weigh military objectives against civilian harm, and adhere to codes of international law. Personally, I'd rather trust human beings with moral decisions, but seeing as we have robots fighting our wars already, putting some safeguards in them is better than nothing.

Proof by blog: Fields medalist mathematician Timothy Gowers decided to run an experiment on his blog by challenging his readers to collaboratively prove a mathematical that he himself could not. Six weeks and hundreds of collaborators later, the theorem was proven, and is planned for publication under the name DHJ Polymath. This success inspired the creation of the polymath project, which aims to advance mathematics through "massively collaborative mathematical research programs".

Conditional microfinance: The website kickstarter.com matches prospective philanthropists with artists, journalists, inventors, and others needing funding for their projects. The twist: unless a project attracts enough funding to meet its needs, no one pays a dime. So you don't need to worry about throwing money at something you're not sure anyone else will invest in; just pledge and see what happens!

SmartTrash Here's a case where I'm not so excited by the invention itself (a garbage can that scans barcodes items as they go in to see if they can be sold for money) as with the general idea it portends: I've always thought of our trash system as one of the worst inefficiencies in our society, in both economical and environmental terms. Outfitting garbage cans with microchips is a possible first step in designing a waste management system that isn't actually wasteful.

Finally, there's one "idea" that involves a complete misunderstanding of evolutionary game theory, as far as I can tell. I'll give this one a separate post when I get around to it.

Book Review: LOGICOMIX

2009-11-23T08:16:00.000-08:00

We are living in an age of, amongst other things, excellent graphic novels. One shining example, which I have just finished reading, is LOGICOMIX, a graphic novel biography of mathematician and philosopher Bertrand Russell. (Side note: can a biography still be called a graphic novel? Our terminology may need an update.)

Seeking an escape from his authoritarian religious upbringing, young Bertrand turned to mathematics as the one source of absolute certainty in his life. But the more he studied mathematics, the more he realized that underlying all the sophisticated theories of the time were arguments based more on intuition than full rigor. Driven by his quest for absolute truth, Russell embarked on a project to rebuild mathematics from the foundations up, and thereby establish its status as absolute truth.

Unfortunately, his project ran into major difficulties of the mathematical/philosophical variety (to say nothing of his equally great personal difficulties) including the famous paradox of Russell's own invention, the arguments of his student Wittigstein that logic was merely a tool for generating tautologies, and finally, Godel's proof that even in the self-consistent world of mathematics, there must always be true statements that cannot be proven.

In the end, though Russell and his contemporaries eventually succeeded in placing mathematics on a rigorous footing, the dream of a logically grounded "universal truth" had to be abandoned. Mathematics is only as true as the assumptions it rests on, and cannot even prove all that is true in its domain.

While the mathematical and philosophical ideas are well-illustrated for a lay audience, the heart of LOGICOMIX is Russell's personal struggle, first to find the universal truths in mathematics and then to accept their nonexistence. Like others engaged in this project, Russell's struggle with logic occasionally veered into a struggle with sanity. Through a meta-narrative of the book's creation, the authors debate the "logic and madness" theme, and ask whether some amount of detachment from reality a prerequisite for one who spends his or her life searching for its foundations.

This narrative of Russell's quest had personal resonance for me: I went through my own late-high-school/early-college phase of viewing mathematics as a bastion of truth in an illogical world. I wonder if many of my mathematical colleagues' careers had their genesis in the same yearning for certainty. I imagine we all eventually come to the same realization as Russell: that mathematics is a powerful tool for clear thinking, but the only "truth" it contains is ultimately tautological.

Disillusioned by his self-described "failure" but ultimately freed from his need for unblemished truth, Russell turns to more worldly concerns, including pacifist activism and the founding of a school with no rules (spoiler: it doesn't go well). The book ends on a bittersweet note as Russell encourages students to accept their lives in an uncertain world.

I had great pleasure following Russell's journey, and the many ideas and people encountered along the way. If anyone is interested in what really drives mathematicians, this book is heartily recommended.

Unsustainable

2009-10-19T11:40:00.000-07:00

The following question was given as a homework problem in a course I'm TAing:

CNBC had an interesting program on the current financial crisis. They located one investor who noticed that since the late 1990's housing prices have been growing 10 percent every year (that is, each year, the average home price is 1.1 times the average price in the previous year) while income was only increasing by 5 percent each year (that is, each year, the average income was only 1.05 times the average of the previous year).

Explain why it is "absolutely clear that this situation could not go on forever", in the words of the investor (who made over a billion dollars because of this observation).

This simple question goes right to the heart of the financial collapse. I would only add that, not only did this particular investor make billions off this observation, but our whole economy lost trillions, because the vast majority of financial decision makers were either unable or unwilling to make this same observation.

(Anyone who needs help with the mathematics of this problem can meet me in the comments.)

Human Cultural Transformation Triggered by Dense Populations

2009-10-12T21:08:00.000-07:00

Biologically,modern humans first appeared 160,000 to 200,000 years ago. But the transition to complex human societies, with art, music, advanced tools, occurred a good deal more recently, and moreover, occured at different times in different parts of the world. An article in June's Science magazine (see a less technical write-up here) argues, based on historical evidence and computer simulations, that in each case the transition was triggered once the population density had reached a critical threshold. At this threshold, there is sufficient interaction to allow for complex ideas to be passed down through generations, enabling rapid cultural evolution.

This highlights an interesting evolutionary tension: as I've written before, evolutionary theory tells us that cooperative behaviors are more likely to evolve (biologically speaking) in populations that are dispersed over space rather than densely packed. But I'm beginning to think that cultural evolution may be different enough from biological evolution to require its own body of theory.

Inferring Social Security Numbers from Birth Data

2009-09-04T08:44:00.000-07:00

An article in July's PNAS investigates the possibility of predicting a person's Social Security number from their birth date and place. Exploiting patterns in how SSN's are assigned, authors Alessandro Acquisti and Ralph Gross developed an algorithm which could correctly predict the first 5 digits of a social security number 44% of the time, for people born after 1988 (older SSNs are significantly harder to predict). The accuracy varied from state to state; for smaller states and recent birthdays, the algorithm could sometimes predict an entire SSN on the first try.

Think you're safe?