“Hitting a baseball is the hardest thing to do in sports.” —Ted Williams
On the subject of noncommutative things from everyday life: what’s the hardest sport? People love to debate this question. Fans with a favourite sport say their athletes are the fastest, strongest, most adept, or otherwise better than athletes from other sports. These disputes se basan the television show Last Man Standing, which pits strongmen against outdoorsmen against finesse athletes against … yoga instructors? Well, I’ve even heard the argument made that skateboarding is the most difficult sport.
One way to justify that one sport is more difficult than another is to measure how long it takes for sportsman A to master sportsman B’s sport and vice-versa.
This gives rise to a connected graph with sports at each node. Suppose an objectively hardest sport exists — i.e., it is easier to transition from Calvinball to another sport than the reverse, for every sport which is not Calvinball.
Even if such a sport exists, there’s no reason to think that the graph would look at all symmetrical, transitive, or obey other nice mathematical properties. addition or composition would be obeyed on the graph. In the scenario I drew above, the ease with which a Calvinballer can transition to another sport tells you very little about how easy it is for an athlete from Sport S to transition to Calvinball. One could measure difficulty of a sport other than Calvinball either by how hard it is for a Calvinballer to transition to it, or by many other measures (aggregate or individual).
It would be more accurate to put individual athletes at each node rather than “a sport”. That I can mathematically write down either scenario shows how varying levels of abstraction (even prejudice) can be incorporated into a mathematical model.