I recently had a very interesting conversation with someone who had a background in computer science and information systems, to see what their perception of math (and mathematicians) was like. It is frequently the case that computer scientists (as well as physicists, biologists, social scientists, and so on) use the mathematical tools relevant to their needs without necessarily having a deep appreciation for the underlying framework that they fit into. This is certainly both reasonable and cost-effective in many cases. After all, unlike mathematicians, these scientists also have to know everything about their own fields, leading some to suggest putting even less emphasis on mathematics, a position I argued against previously. However, I often get questions like the one recently popular on Quora: What is it like to understand advanced mathematics? I often get the related question: What is it like to use it?
When I try to answer this question, I usually break the answer down into three stages. The first one is exploration, which involves a lot of trial and error. The second one is analysis, which comes from systematically thinking about your experiences. The third one is confirmation, which allows you to make sure that what you are doing is right. Let me try to briefly explain each one of these on an example that comes from my doctoral work on regulatory networks – it is simple enough to follow without much background, yet complex enough to enter unknown territory.
In graphs (collection of edges, which you can think of as roads, connecting vertices, which you can think of intersections), one common problem is to rewire the edges in a way that preserves the degrees (the number of edges found at every node). For example, one would like to get from the graph on the left to the graph on the right.
If the edges (roads) don’t have directions, it is known that switching a pair of edges (that is, going from the configuration on the left to the configuration on the right) is enough to get from any graph to any other graph that preserves the degrees.
However, if the edges have a direction (that is, a beginning and an end), and we need to preserve the number of incoming and outgoing edges at each vertex, then the switches are not enough, and so-called triangle flips are needed, as shown below.
The problem I was facing was one where each edge not only had a direction, but also one of two types, say “solid” or “dashed”. What were the moves that I had to allow in order to get from a given graph to any other graph with the same number of incoming edges of each type and the same number of outgoing edges of each type at every vertex?
The exploration stage started with some simple examples, like the ones below (with the goal of obtaining the same configuration, but with edge types reversed), where a simple sequence of edge switches or triangle flips would not be enough.
The discovery stage involved trying to see how those “obstacles” could be fixed. Pretty soon I discovered a procedure that actually allowed me to do this, provided I had some extra edges that were available elsewhere in the graph. For instance, the picture below shows how to deal with the configuration on the right (which we call a strong triangle) in 4 steps, using three extra edges. First, switch each triangle edge with an extra edge. Second, flip the problematic triangle. Third, switch the edges to restore the parallel triangle. Fourth, switch once again to restore the auxiliary edges.
The confirmation stage was to see whether it was enough to be able to fix these bad configurations, or if there were others that could not be fixed with this approach. Unfortunately, as the computer simulations written by my co-author showed, there were other configurations that could arise that did not yield to our techniques (even though we never encountered any in practice). For this reason we left this problem open for whoever was going to be interested in pursuing it further (it is still open).
This process is more or less what happens whenever I try to solve a mathematical problem. If we compare it to hiking, then the exploration stage is like getting your bearings; the discovery stage is like observing the terrain and making notes; the confirmation stage is like making a map that others can follow to retrace your steps. But what exactly makes the discovery happen? This will be the subject of a future post!
P.S. I’d like to thank Amy Rossman, Wikipedia and Google Image search for the images used.