Inside the Mathematician's Studio

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

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?

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

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

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

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?

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.