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

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

Now lets assume there is some process for picking out universes from the multiverse.  Since there’s no time in the multiverse, the process has no beginning and no end.  It’s like a computer program, but it’s infinitely complex.  Let’s call it The Process.

If The Process is infinitely complex and has no beginning and no end, what can we know about it?  We know that it picks some universes but not others, which effectively creates an “in group” (all those that are picked) and an “out group” (all those that are not).  Of course, both sets are infinite and still have no structure.  But note that all the universes in one group or the other now stand in relation to one another.  That is, they share the property of “in-ness” or “out-ness”, and between the two groups there’s the relationship of “different”.

The Process further divides these sub-multiverses in unknown ways, and this sorting creates other relationships between universes.  You can visualize a network of universes with the connections representing these relationships.  The network is infinite, and if you consider any subset of the network, it’s also infinite.  But these subnetworks are no longer arbitrary, they are networks themselves and networks have structure.  And since a subnetwork by definition shares the same connection relationships as the original network it is a “sub” of, the subnetwork is structurally similar to the network itself.  That is, the network is self-similar, which in mathematical terms means it’s fractal.  Of course this fractal we are talking about is infinite, and so wherever you start, it’s turtles all the way down, and all the way up.

Notice that the process of identifying subnetworks does something interesting, it creates an asymmetry that wasn’t distinguishable before.  For any network N, if you choose a subnetwork, n, then N “contains” n but not vice versa.  This containment relationship can viewed as a network where the connections are arrows, meaning they have directionality, N –> n.  You may have noticed that we just went from talking about a network of universes to a network of networks (of universes), but that’s okay.  Remember the multiverse is infinitely infinite, and we’re just chatting about some arbitrary aspects of it.  There’s lots of other aspects we could talk about instead, but it’s starting to get interesting here, so let’s continue….

Somewhere in the fractal multiverse network of networks described by The Process there is a subnetwork (actually an infinite number of them) where the structure is like this: each universe is connected to by only one other universe but connects to an infinite number.  Let’s call this structure, Time, and note that there are an infinity of subnetworks of the network which have this Time structure.  Unless stated otherwise, I’ll be talking from now on about networks with Time structure.

Remember though, the multiverse itself has no structure; The Process overlays structure on top of it and thereby allows us to know about things like Time.

Now let’s start using the words network, system, particle, entity, agent and universe interchangeably, so we can say things like “time network”, “temporal system”, “particles over time”, and “A causes B” to refer to roughly the same thing.  I realize that by overloading these terms I’m jacking into (and hopefully hijacking) your intuition about what these words mean, but that’s my intent.  Hopefully you’ll continue playing along by my rules and try not to project what you already know onto this alternative cosmology.

When we use the words network, system, particle, entity and agent, you might wonder whether we are talking about a universe or a multiverse.  The answer is Yes.   Remember, the multiverse is infinitely infinite and self-similar, so in some sense we can say it contains itself.  We have a hard time with infinity so this concept is mind-boggling, but if you follow the logic, hopefully you’ll accept this paradox as true.  So lets just use the word universe from now on and forget about multiverses.  And to not get confused, let’s refer to what we used to think of as the Universe as the known universe instead.  The known universe is where you live (or more precisely where you think you live) along with everyone and everything you know about or can imagine.

The known universe is expanding the more you learn about it.  The known universe is temporal.  And as we know from Einstein, it must therefore also be spatial — remember it’s not space and time but rather the spacetime continuum.  The known universe consists of particles (i.e. matter) and therefore — also thanks to Einstein — it consists of energy.  Time, space, matter and energy here may or may not be totally in sync with our intuitions of them, but just suppose they are the same thing and that our intuition is slightly biased by our particular experiences in life and could use adjustment.

We haven’t really talked explicitly about laws of nature, fundamental constants, invariant equations or even mathematics.  And I kinda jumped the gun when bringing Einstein into the equation (so to speak).  But it’s really hard to follow a line of thought without some sort of logical paradigm, some structure of thought.  In the end it doesn’t really matter what I’m saying, what you’re hearing, or whether any of this is “true”.  I’m just telling you a story, and hopefully it’s amusing enough for you to finish reading.

Originally we talked about The Process, which is infinitely complex and which describes all sorts of possible realities.  The known universe is one of those possibilities, one in which we see structure and patterns, order and complexity all around us.  Somewhere “out there” there may be portions of the multiverse (whoops, I said I wasn’t going to use that term anymore, sorry) where it’s still appears, unstructured and thus unknowable.  But let’s come back to the known universe and the “knowable” universe.

Because of the fact that we are here in the known universe thinking and talking about it, and not in some unknown or unknowable part, the non-random patterns that we see may look to us like universal laws (E=mc^2, the second law of thermodynamics, etc.)  Well, we know that even these laws are not truly universal, they apply to only certain scopes.   For example, “relativistic but not classical or quantum realms”, or “closed systems but not open systems.”  String theorists are looking for universal laws, but so far none have been found.  But let’s just grant them that they will eventually find some (or one).  How would we be able to distinguish between a true Law and just a pattern that is very very persistent over all known scopes?

How about we stop using the word “law” and instead replace it with the word “principle” to suggest that it may really just be a pattern that we see in the known universe.  And as the known universe expands via our increase in knowledge/understanding/awareness, we might find exceptions to the pattern.  After all, that’s what’s happened to every “law” ever considered in the history of science so far, and why should that pattern stop?  (Sorry, my paradox detector just went off, let me reset it….)

Coming back to principles, there’s one that emerged from the last few paragraphs, did you notice it?  Cosmologists call it the Anthropic Principle, which is the notion that the universe appears ordered in the particular way that it does with these nifty laws and constants because of the very cosmic coincidence that we are here observing it!  In other words, we live (and can only live) in the known universe, by definition.  And we wouldn’t be “here” and able to “notice” anything if we were in some unknowable part.  That’s a pretty trippy concept, but one that many physicists take very seriously.  It’s the same kind of argument as for why we haven’t been contacted by aliens yet: there’s a decent chance we are the most advanced intelligence out there and we’ll have to wait for others to catch up so we can communicate.  It’s also the reason that your keys are always in the last place you look.

Remember the Anthropic Principle because it’s really useful.  It has the same logical structure as Darwinian evolution and other “emergent” phenomena.  Is this Generalized Anthropic Principal (GAP) a universal/fundamental one?  Who knows.  Probably not.  We anthropic agents are so self-absorbed.

Another principle that emerges from our cosmology is Coherence.  Because of The Process, birds of a feather flock together.  Actually, The Process defines which birds are of which feather, so this is a tautology, though it’s fun to think of it as “like attracts like”.  But we know that really it’s just co-incidence: the birds exist at the same Time.  Using the analogy of birds, we can ask whether these coincident birds are different birds or the same bird.  But it’s a silly question because the answer is Yes.  Think of quantum coherence, if you like.

Now let’s say we are talking about particles and not birds, and instead of Coherence we’ll say Gravity.  Isn’t it the same thing?  We talk about stars and planets and other astral bodies as if they were coherent entities, but If there were no gravity, would those entities exist?  Or let’s talk about the Cooperation of the cells in your body; without it, would you exist?  We’ve all heard about the “law of attraction” from The Secret, isn’t it the same thing?  You imagine the future you want, and that acts as a beacon guiding you in every decision you make, every micro-decision, every unconscious action until at some point you find yourself living in the future you imagined.  Coherence, cooperation, attraction, unity.  Same thing.

Here’s a secret: there is no Process.  Or if you prefer, The Process is completely random.  Yet that doesn’t change anything I’ve said above.  Think of it this way: in an infinite series of random numbers, all patterns appear eventually, right?  So “somewhere” in the infinite randomness, The Process “produces” the structure I’ve been talking about.  Or maybe the fact that we’re anthropically talking about it produces the structure.  We are The Process.  Or more generally, we humans are part of The Process.  The Process is the universe.

Related posts:

1. Why Falsifiability is Insufficient for Scientific Reasoning
2. Notes from TED
3. Synthesis of Complexity 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.

Related posts:

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.)

Related posts:

1. Name That Financial Debacle!
2. Radical Transparency
3. If Rafe Were In Charge: Major Medical Edition