“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. Evolutionary Game Theory and Archaeology
3. Game Theory and Military Planning

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.

No related posts.

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. The Idea of Applied Mathematics

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. Evolutionary Game Theory and Archaeology

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. What is a Gene?
2. Evolution & Emergence
3. Evolution Favors Cooperation Over Competition

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

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!

Related posts:

1. Game Theory and Military Planning
2. The Future of Evolutionary Theory?
3. Generalized Evolutionary Theory

• 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.

Related posts:

1. The Challenge
2. X Prize Annuity Funds
3. Truthocracy – Part III – MIT Center for Collective Intelligence

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.

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1. Highlights from the Year in Ideas

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.)

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1. Name That Financial Debacle!
2. Radical Transparency
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