Science Corner: Multidimensional Justice

We just observed Super Bowl Sunday, the culmination of the American football season. There are downsides to having so much of our attention and our culture revolve around the sport. On the plus side, it provides common ground for metaphors. One of the quirks of football is the possibility of offsetting penalties. I’m not sure if we saw them in this Super Bowl; I didn’t notice any, but the slow-starting game lost my attention for a while. But if you’ve watched at least a few football games, you’ve likely come across them. Basically, if a player from each team breaks the rules on the same play and the infractions for the penalties are similar enough, they cancel each other out and neither team is penalized.

Matters of justice have dominated national conversations of late. How do we implement justice for criminals and the victims of their crimes while respecting the rights of the innocent? How do we balance justice for breaking immigration laws with compassion for those seeking to escape difficult circumstances not of their making? How do we hold the powerful accountable for crimes against the most vulnerable when the powerful control so many of the levers of justice? Answering all of these questions is beyond my remit, but just because I am treating them rhetorically does not mean I think they are unanswerable. Rather, I want to focus on what I think is a mathematical aspect of those conversations.

I had offsetting penalties on the mind even prior to the Super Bowl, because the idea feels like it is creeping into some of these other, more consequential matters of justice. Maybe not to the same degree in all issues, and certainly not in everyone’s reasoning, but I have noticed that offsetting mindset at play. Someone you identify with, respect or support is legally or morally culpable? Just find someone from “the other side” who is similarly guilty. Then your proxy, and by association you, are off the hook, right?

Well, why does that work in football? The nature of the game means that you can collapse much of the action into a single dimension–distance between the ball and the goal line. The discrete nature of the action also helps, with the ball not moving between plays. That allows officials to adjust the position of the ball in that one dimension while action is paused, as a way to punish rule violations. So when multiple infractions would mean shifting the ball in both directions, you might as well just skip ahead to the net zero movement. But note that this is not common across all sports. In ice hockey, if players on opposite teams commit penalties, they are both sent to the penalty box rather than allowing those penalties to cancel out. Same thing with yellow or red cards in soccer; you don’t get away with something just because I also broke the rules.

In a sense, this is because the punishments occur in separate dimensions. My time in the penalty box is not on the same “axis” as your time in the penalty box, such that any shift towards more time for me means less time for you. I doubt the officiating bodies of these sports necessarily think in terms of dimensions of punishment when constructing the rules. But that’s one of the joys of math, finding ways to apply common abstractions to a variety of situations so that reasoning and problem-solving can be shared between them. I suppose one could say something similar about sports and other games; they provide abstractions of competition and conflict in which we can work out different approaches with lower stakes.

Of course, life can’t be all fun and games. The stakes in the questions I raised earlier couldn’t be higher for many involved. All the more reason, then, to explore the full dimensionality of the possibilities for justice. So let us note that something interesting happens in high dimensions. In one or a few dimensions, there is a strong sense that some points are closer together than others. My son is less than 300 miles away from me at college; my daughter is over 2,000 miles away. Those aren’t the same distance, and it matters; I can drive to one in a day but not the other. But in high dimensions, most points wind up roughly the same distance apart–not necessarily close, just similar in distance. This creates challenges for data analysis known as the curse of dimensionality, but for our purposes now let’s consider how it might be a blessing.

To even be able to contemplate offsetting justice, I had to introduce the idea of someone from “the other side.” Notice how that is also a one dimensional concept, assuming we can put everyone along a single axis and split them into sides. This assumption is front and center of much public discourse at the moment. As a result, we naturally regard folks as either close to us or far away from us in that one dimension. Consider then what would happen if we instead regarded our fellow humans in their full dimensionality. As we expand into more dimensions, the distances between us smooth out. Again, they don’t go to zero. We are definitely not all the same person, and we don’t need to be for this effect to work. Quite the opposite, really. Since we all contain multitudes, between any two people there will be enough of a mix of commonalities and distinctives to put a similar distance between them as between any other pair.

Putting it another way, John Hodgman likes to say “Specificity is the soul of narrative.” It is his way of encouraging guests on his podcast to be concrete and detailed. And he’s expressing a sentiment many have observed about storytelling, that universal stories require specific details, not generalities. This could seem like a paradox, but it makes sense if one reflects that specific stories are high dimensional stories while vague ones are not. And so a vague story will perhaps resonate with some strongly, but potentially feel quite alien to others. But a story with many specific facets will be similarly relatable for nearly everyone.

So take this as encouragement to be fully yourself in all the dimensions you occupy. And appreciate the fullness of all of God’s other image bearers as you encounter them and consider how to treat them justly.