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.

Related posts:

1. Game Theory and Military Planning
2. Evolutionary Game Theory and Archaeology
3. What Obama Needs to Do

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, 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 principles.

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.

Related posts:

1. Decision Education Foundation
2. Education 2.0
3. A Middle/High School That Teaches Complex Systems Through Games??!

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.

Related posts:

1. The Logical Necessity of Group Selection
2. Evolution Favors Cooperation Over Competition
3. Generalized Evolutionary Theory

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.

Related posts:

1. Gene-culture Co-evolution
2. Symbolic Representation is the Key to Major Evolutionary Transitions?
3. Human Cultural Transformation Triggered by Dense Populations

“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.

Related posts:

1. A Middle/High School That Teaches Complex Systems Through Games??!
2. Education 2.0
3. Evolutionary Game Theory and Archaeology

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.

Related posts:

1. Universal Constants

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.

Related posts:

1. Evolutionary Game Theory and Archaeology
2. Generalized Evolutionary Theory
3. Symbolic Representation is the Key to Major Evolutionary Transitions?

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.

Related posts:

1. Generalized Evolutionary Theory
2. The Future of Evolutionary Theory?
3. Is a new mode of evolution emerging?

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?

Related posts:

1. Is a new mode of evolution emerging?
2. What is a Gene?
3. Evolution & Emergence

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.

Related posts:

1. The Future of Evolutionary Theory?
2. This Sentence is False
3. Complex Systems Concept Summary