Mathematicians and Ant Trails

Ant colonies perform an interesting procedure for route optimization. Many ants will wander around looking for a new “score”; a source of food, all the while leaving weak pheromone trails When an ant finds some delicious dropped ice cream, it follows its pheromone trail back to the colony and tells all of its friends. These friends follow the trail back… but not perfectly! They try to opportunistically take shortcuts. Over time, faster paths get found, and the original trail fades in preference to more efficient routes.

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This is the inspiration for a family of algorithms called Ant colony optimization algorithms. But the exact details of this algorithm are not what I want to talk about now.

What I want to talk about is that I think a very analogous process happens in mathematics. Mathematicians wander around in conceptual space until they hit some metaphorical smashed watermelon, and then they tell all of their friends the path that they took to get there. For instance, they might have been trying to solve some specific problem, like showing that there exist polynomials of degree 5 whose solutions cannot be expressed with radicals, and stumbled across a useful theory; the theory of groups. But their friends don’t always listen too carefully, and look for their own paths to the watermelon, and this process repeats, potentially forever, as better and better paths are found. Now we use groups for all sorts of things that are not Galois theory, and moreover we’ve found much simpler ways to teach groups than the way that they were discovered. This was a result of many “ants” lossily retracing the steps of the original ant, and making the path clearer.

I have two points to make about this. The first is that academic mathematical culture gives a disproportionate reward/prestige to the first ant, the ant with the complicated pheromone trail that initially found the half-eaten cake, and not enough recognition to the other ants which make the path simpler.

The second is that category theory could be viewed as a powerful technology for cleaning up ant trails. I.e. just the process of taking a mathematical discovery and rephrasing it in categorical language does so much to clarify what parts of the original trail were necessary and what parts were accidental.

These points are connected; I believe the way that some mathematicians feel about category theory, that it isn’t “real math” in the way that solving open problems in number theory or combinatorics is, is connected to the attitude that reshaping ant trails is not valuable work.

I’d like to end on another note though. In general, it’s a good thing that ant trails get simplified. But often the original ant, in their twisty ways, found some other tidbits that were forgotten in the push to the main event. And the skill of forging new paths is different from the skill of following well-trodden paths; seeing the decisions made by those original ants can be quite illuminating. So it’s important to both value the original path, and value the work put in to turn it from a dangerous trek through the wilderness into a comfortable high-speed train corridor.