Below, screenshot from the game , a "playable theory" -- loosely based on Jullian Assange's on the apologetics of information transparency.

From the same series we have strategy game (for aspiring State Department employees one guesses) and the spare and haunting game about, well, the genetically engineered end-of-the-world (the music is especially beautiful). To top it off the Catholic Church and also come in for some playable satire.

( download )

I've been dwelling on the kinds of intuitions developed by anyone who studies mathematics. They're often simple rules of thumbs and ways of thinking about mathematical objects. In fact, they are so obvious that once you've internalized them, it often doesn't occur to you to articulate them again.

For example, one very basic component of real analysis is function composition , something that is probably taught early on in high school -- although .

What does it mean to compose functions? How does one reason about a compound function? I got thinking about how one might go about helping a student to develop their intuition about these questions. It occurred to me that there is a simple visualization technique that answers this exact question without any words at all .

How does this work? Let's say you want to visualize the following compound function:

First, let's consider the syntax tree of this expression. This is a tree in which the root is the entire expression and the leaves are linear or constant functions like x and 1/3 . Luckily, with Mathematica it's pretty easy to present an arbitrary expression directly in this tree representation by using the formatting construct TreeForm :


My idea is to produce actual function plots for each interesting node in this tree. By moving up the tree we can show how these sub-expressions fit together to compose the entire expression . It turns out that it takes about 15 lines of Mathematica to compile the syntax tree, recognize and extract the interesting nodes, and synthesize the corresponding plots into a graphical diagram.

Wrapping this all up into a function called FunctionTreePlot , we can now visualize our example like so:


This technique seems to work quite well. You can easily chase visual features of the corresponding plots up and down the tree to answer questions like "why does this function have a pole here" or "what will be the effect of changing this co-efficient?" No doubt this functionality would be a great addittion to Wolfram|Alpha's already strong support for visualizing mathematical functions .

Here are a few more examples from my experiments:

I recently completed a week-long project to analyze the performance of today's dominant search engines when presented with questions from the famous Jeopardy! game show. The originator of the idea, Stephen Wolfram, used the results in a blog post about the similarities and differences between (where I work), and IBM's intriguing new question-answering system .

If you have already seen one of IBM's television spots, you'll know that in mid-February the system (dubbed "Watson") will compete on a special episode of Jeopardy against the two top Jeopardy champions Ken Jennings and Brad Rutter.

This event is likely to go down as an iconic example of the advance of AI technology into a realm previously reserved for human judgement, a touchstone that is similar in many ways to IBM's successful challenge to the reigning chess world champion with its computer in the late 90s.

For my part, I looked at how well traditional search engines allow one to narrow down the vast corpus of online information to just a page of potential answers to a Jeopardy clue. Not badly, it turns out, although Watson will surely advance the state of the art in text-corpus question answering.

You can find more information on , but I'll reproduce the main bar chart here, along with a fun little word-cloud I made (with the help of ) of the types of entities that occur as the answers to roughly 200k Jeopardy clues. For one thing, it's interesting how close all the major engines are now becoming, as powerful web search increasingly becomes a commodity we take for granted.

( download )

The inventor of , probably also one of the greatest mathematicians of the 20th century, was a rather astounding character named . When he wasn't throwing fabulous parties in Princeton or working on the atomic bomb, Von Neumann was laying some of the groundwork for the science of the late 20th century. Among his many creations was one of the first electronic digital computers, and associated with it the stored-program architecture that still underlies computers today.

A slightly less well known creation of his was the so-called . Something of an alter ego to the digital computer, the cellular automaton is a distributed computational device composed of an array of many simple processors instead of , as in the computer. Unlike Neumann's other ideas, which continued to flourish, cellular automata fell into obscurity until the late 70s, when they experienced something of a renaissance after the of (who happens to be the CEO of the I work for).
More recently, a strange blend of these two ideas has arisen in the guise of rather long-winded . Various academics have what happens when, instead of the disembodied entities of traditional game theory (who one imagines lurking in imaginary war rooms plotting ), the game players are embodied agents situated in physical space on a grid or lattice, interacting with their neighbors.

It turns out that this can make a huge difference to the dynamics of such games. For example, in the traditional game of Prisoner's Dilemma , the only evolutionarily stable strategy (a strategy that a group of agents can play without being undermined by the appearance of mutants in their midst) is for every player to defect -- in other words, a lose-lose situation where everyone mistrusts everyone and we are all unhappy. This is nature red in tooth and claw, pre- Leviathan .

However, when we consider that these agents can be on a grid, and can, so to speak, huddle together for comfort, we notice that "tribes" of co-operators can form that offset the attacks from defectors on all sides by forming many positive relationships amongst themselves (see pictures below, the top panel is where the co-operators (blue) have successfully fought off invaders, in the bottom not so much). In fact, this point gets to the heart of one of the puzzles of evolutionary biology: why altruism exists and how it evolved.

This last result was discovered only recently with the help of computer experiments, and there are sure to be many such discoveries waiting to be made in the field of evolutionary spatial game theory. To this end (yup, all of this post so far has been background -- it links up too beautifully to not mention the history), I've been working on a software library to allow experiments in ESGT to be conducted elegantly and efficiently. To leave you with a taste of what such experiments to look like, here are some videos of the game of Rock Paper Scissors being played on grid (at various 'temperatures', or degrees of randomness):

I'll post more such videos in the future as I do further experiments, and hopefully delve a little more into the details of the science that is going on. If you're interested in the code and seeing some more examples, or even playing around with it yourself if you have a copy of Mathematica , you can visit the project page on GitHub:

Long live spatial game theory!

Today, a science lesson delivered via Mathematica .

Imagine a pawn on a chessboard. Instead of marching steadily forth, this pawn has gone renegade. In fact, it has gone completely mad, and without respect to the rules, is shuttling randomly forward and backward. Every time it is your turn, it is equally likely to:
1) not move at all,
2) move forward,
3) move backward.

Such a monstrously aimless behavior is called in mathematics a .
Here is some Mathematica code to simulate a random walk, and to collect the statistics of many such simulations.

What does a random walk look like? If time is vertical and the progress of the random walker is horizontal, one particle performing 50 steps of a random walk looks something like this:

What about if one superimposes many such walks ?

As you can see, with 5000 particles, the identity of each particle has begun to blur away, and now the whole ensemble starts to look a little bit more like a blob of ink diffusing on blotting paper.

This is not an accident . That is exactly what ink diffusing on blotting paper is: a near infinite number of molecules of ink, each one buffeted hither and thither by thermal vibrations, each performing what is called in the biz .

In fact, one of Einstein's achievements (other than the various flavors of relativity, of course) was to show mathematically how this Brownian motion relates to the underlying atomic nature of matter.

A garden path musing leading to nothing

What is postmodernism? I don't really know, but I have a feeling about what it is. This feeling comes from one or two books, many conversations with students of philosophy, an article here and there, and especially being privy to the torrential backlash from strident rationalists and their op-ed pieces in Skeptic or Scientific American . I used to openly mock it, but now I have a different idea of its value.

The last few months I have been, largely unintentionally, building a new theory of the world. Theory isn't quite the term: a loose coalition of theories, a web of many strands that I periodically find an opportunity to expand outward or knit inward. What sort of theories? Theories of how to think, how to identify myself, what identity is, what it is to be human, what technology and culture are and how they shape us, how we relate to each other and the world, what spirituality really means, what knowledge is, and the meaning of mystery. And all of them quite uniquely flavored by my history -- and strangely, this subjectivity comforts me.

And the theories themselves aren't strident recipes for understanding. Indeed to have any theories at all about such high-level constructs one has to abandon the luxury of empiricism or even the presence of a coherent argument. But that is the price you pay for leaving the world of objects and agents and entering the world of meaning, people, beliefs, and culture. And, while it takes faith in your intelligence and your wisdom, and it tests your tolerance of ambiguity and paradox, the rewards are manifold.

It seems to me that of the many things we become emotionally and intellectually invested in -- religion, politics, art, science, literature, sport, society, whatever they are -- most serve as hotbeds of proxy symbolism, as foundations on which to bolt the meaning of our lives, and so they unavoidably trap us. And then, trapped by our existential need for these things, we are subsumed by the detail, distracted by the texture. Instead of being porous to the experience of life, the tide of mystery goes out and we become like fetid pools, gradually growing more uncomfortable.

What is real life? There is a mystical quality to it, I believe: the acknowledgement that nothing really makes sense, and furthermore, cannot be expected to make sense. The world is flux, chaos, contradiction; but these qualities only scare us for as long as we expect the world to be rigid, ordered, amenable to the rather pitiful analysis that we, as humans, are equipped to perform.

So how to respond? I guess laughter is important. Laugh at what you cannot understand; laugh at what you think you do understand, because you are surely wrong; and especially, laugh at others when they realize they have no idea either. And so too with empathy: give in to the welling up of sorrow, sadness about the violence and harshness of the world. But also delight at its small and plentiful daily gifts, for they will never dry up.

Okay, I now realize that I've sort of lost track of what I wanted to say. But maybe that is quite appropriate.