The Idea of Applied Mathematics

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 insane, 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

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

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

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

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

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

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?