Or, Why Studying Math Is The Best Thing You Can Do
In the hallway the Academy of Management conference last week I ran into a colleague who mentioned that he’s been following CRTVTY (hurray!). But he was surprised. “You’re a math guy,” he said. “What’s a math guy doing with a blog on creativity?” Well, aside from all of the research linking math to music and art, this brings up an interesting question: can math itself be a creative exercise?
Before we dive into that question, we need to clarify what kind of math we’re actually talking about. Those terror-inducing sheets of three digit addition problems from second grade don’t make candidates for creativity, but their big brother might. My research relies heavily on a tool called mathematical modeling, which basically involves boiling a real world phenomenon down into a set of equations that can be used to gain insight into the initial problem. In this context, a model isn’t unlike a dollhouse or a model airplane: it’s a simplified representation of an otherwise inaccessible system. The classic supply and demand graph from your intro econ class is a fantastic example. This model is a drastic simplification of how markets really work. In reality, there aren’t smooth lines that describe how producers and consumers go about their business, an apple today isn’t equivalent to an apple tomorrow, and people aren’t always (or ever) rational in their decision-making. At the same time, accounting for all of those complexities really isn’t possible, and the two lines actually provide a lot of insight into things like how prices rise and fall, and what products people decide to buy. Thus, some realism is sacrificed in order to make the problem more tractable. My research, for example, explores how firms collaborate with one another in order to build high-tech products. Models like mine thus consist of sets of equations that describe a range of actions that firms can take, and how they benefit from doing so. By solving these equations, I’m able to understand the issues that shape firm strategy under various conditions. It isn’t perfectly realistic, but it allows a lot of insight into an issue that’s otherwise unintuitive and difficult to untangle.
So how do we build these models? Well, the first step is to understand the phenomenon being modeled. In this sense, building a model isn’t unlike painting a portrait: easy to do, but better when you actually know what your subject looks like. This theoretical understanding is then translated into math, which could look like anything from a handful of hand-written equations to a complex computer simulation written in Java or Matlab. Once you’ve got a model, you then analyze it to see if you learn anything new. While the approach isn’t without its limitations, modeling can be a powerful tool for understanding complex problems or for exploring tricky research settings. Models of the global economy are used to test and predict the impact of new tax plans, for example, since the “pass some legislation and see what happens” approach isn’t always as viable.
But what on Earth makes this a creative process? Well, just as the painter asks how she can best express herself, the modeler asks how she can best express a problem. The standards by which we judge artistic output – novelty, beauty, relevance – are exactly what differentiate good models from bad. A model should be able to produce novel insight, for instance, and should represent its subject simply and elegantly. Doing so requires the insight to be able to reduce a phenomenon to its core principles, and the creativity to effectively represent those principles within a set of equations. When I was young, my father once described computer programming as the ultimate creative experience; building freely and translating vision into reality without the constraints imposed by the material world. So too with modeling.
Which leaves one final question: why write about it? I believe that viewing mathematical modeling as an exercise in creativity provides three critical lessons. First, modeling isn’t a science, and it isn’t an art. It’s a bit of both, and all of the tools and constraints and peculiarities of the creative process apply. When people get angry about the economic model being broken they’re missing the point: an unhelpful model isn’t wrong, it’s just the wrong choice for what you’re trying to do. Second, creative expression requires having the right tools, be it a sculptor’s knife or differential equations. Math isn’t considered a creative area, and many people flee as a result. This is a tremendous loss: creativity doesn’t exist without the tools to call it to life. Finally, while many agree that math underpins many more traditional creative endeavors like music and painting, few argue that math itself is a creative endeavor. And that means that if we look for creativity only in paint and pictures, we’ll be looking in all the wrong places.