I think a lot about distance. It's the example of a phenomenon that seems intuitive but which can be ineffable. Just reading "distance" probably invokes notions of a particular type of measure, say, the distance from work to home--even if those terms just refer to rooms in your apartment or house. But there are types of distance that rely on, or imply, contexts that taken together can uncover interesting relationships. That's what I want to dig into.
I think there's something in the Magna Carta that requires anyone talking about even tangentially mathematical topics to really lean into modeling the existential apparatus in LaTeX and Mathematica animations. I'm going to defy both this statute and my own inclinations, not only because it sacrifices clarity in an effort to look cool (I said "an effort", okay?), but also because it attributes primacy to a study of things, that being mathematics, over the things themselves. Of course I can only mean "the things as we can perceive them" which takes as given that there is no primacy within our purview, but I can't keep going down this road because I just used "purview" and nothing good will come of it.
So, distance. I brightened when in my first analysis (think "calculus" but fancier) class in college the professor introduced "distance" as if it were novel. This was in service of introducing us to the idea of a metric space, which excites me in the way that conspiracy theories excite QAnonomists: it's a simple idea that can lead to powerful ideas that make sense of the world as seen in a particular perspective. A metric space is a set of elements and a distance function that acts upon the elements. Our common usage is basically the world as a multidimensional set of points and the Euclidean distance function. That is, coordinates in space and the straight-line distance between them, "as the crow flies". It's very useful for estimating the cost of carpeting your home, or relating weather patterns with how far the planet is from the Sun.
There are lots of situations, though, which we assume hew to this Euclidean model but which actually don't. Some of them are close enough to be useful approximations, like, whether Australia is 8,800 miles or 12,000 miles from Colorado, either is good enough for conversation. You could just say, "It's really far away," too. But think about throwing a football for some reason: if you wanted to try to throw a football from one end zone to another of an NFL-regulation football field, the football itself would travel more than 100 yards. The trajectory traces an arc that has both a horizontal and a vertical component, and while the horizontal component sure is 100 yards, the arc length might be like 120 yards. In conversation, and especially as the throw relates to the rules of the game, if for some reason you talk about this you'd say it was a 100-yard pass. If you're doing the throwing, the angle with which you throw dramatically affects the length of the arc the football follows, and so too the chances you'll make it. So at the very least: context matters.
Think about driving to the grocery store. The closest grocery store to me is a little over five crow miles away, but unless the near future holds some real exciting developments I'm not going to fly there. I'm going to drive, or if I were enterprising maybe ride my bike. Each method of transportation will use roads and given there is no unerringly flat and straight ride from my house to the store, the trip will cover more than five miles of road. So, how far away is the store? Or, in other words, what is the distance function for this context? We can think of "as-the-crow-flies" distance as "simple Euclidean" distance, to differentiate it from what is often called Manhattan distance. Manhattan distance is the sum of distances traveled along the relevant axes, here x and y, which you can think of as a grid of streets in a big city. As an approximation, Manhattan distance is a much better model for how far away the Doritos are from my hungry maw, because it factors in some idea of constraints on movement, on crossing the distance.
One of the interesting promises of the mathematics of metric spaces and matrix algebra and the like is the applicability of spatial intuition, even if just as a toe in the water. Manhattan distance uses the shape of hypothetical city street grids as a familiar reference in the casting of deeper magics, for instance. Reading the Wikipedia entry, I'm learning that not only does it help us understand the difference between a straight-line distance and a more-practical distance between two points in a downtown area, but it also can help folks understand RNA splicing. We build the mental wireframe of getting from the Chiquita Building to the Hamilton County Public Library in Cincinnati, say, and superimpose relationships between nucleotides.
Ever played a video game, or even watched YouTube videos about them? You'd have a difficult time not coming across a scripted cleric walking around a fantasy town running into things, or a pack of wolves utterly incapable of crossing a creek to attack you, or similar. Computer-controlled characters in video games move around according to a system of rules usually called pathfinding. There are lots of algorithms for managing this, with one of the most well-known called A* ("A star"). This method represents distinct points of the game world as vertices in a graph, with edges connecting them. The vertices, the points that map the game world, are often assigned weights, some notion of the difficulty of moving to or through that place in the world. What A* and other algorithms do is find the collection of points between some starting point and some goal point, passing through each vertex on its way and calculates the relative difficulty of getting to the goal along a given path.
We do the same things in real life. I recently rode in my first bike race. The course I signed up for was just shy of 40 miles long, and this featured prominently in my assessment of the difficulty. "I've ridden more than 40 miles at a go before, plenty of times," I thought. Of course there was some elevation gain, too, but I've ridden hilly roads before. So I rode some trails and some hilly roads by my house and thought I was prepared. And while I finished the race, I did not finish well. I didn't weight the points of the path appropriately.
One thing I should've considered is that while I've ridden hilly stuff, I've never been a strong climber, and the amount of climbing on this ride was more than I've done before. For another, the race course included paved sections, but also lots of dirt sections of varying degrees of surface conditions as well as a fairly long section of somewhat technical (e.g., rocky, chunky, rutted, etc.) stuff with a slight-to-moderate incline. In the end, I rode 40 miles, sure, but on average each mile was harder to ride than most miles I've previously ridden. They were harder to pedal through due to the dirt and rocks, and they were harder to climb. In a race like this, some notional pathfinding would be tuned to find more, not less, difficult routes. In terms of metric spaces, the distance function here would include some representation not only of the Euclidean distances traveled but effort required to travel them.
A bike race like that is an example of a distance function used creatively, in that case to give riders a challenge. Another kind of distance function I really like is used in horror movies. In a horror context, we often find a protagonist could be very close to safety in strictly Euclidean terms--for instance, outside a door or in a copse of trees 30 feet from help--but very far away in a more comprehensive sense--like their legs are injured making crossing the 30 feet tough, or their voice is hoarse so it's hard to call out for help. Lots of narrative tension is built like this in horror movies and shows. For instance, in later episodes of season one of Stranger Things, Will Byers was in a sense near his mother but he was separated by some kind of dimensional barrier. The dimensional barrier made crossing a very short distance near impossible.
The fears you don't face become your limits.
These are mostly trivial types of distances. There are types of distances people encounter with high variability which have material effects on their lives. Think about that quote: "The fears you don't face become your limits." One way to read this has to do with something I like to (unoriginally) call a solution space, the set of all possible solutions to one or more problems in a given context. Relative to a particular skill or activity, you could think of a metric space defined with you at its center and all possible solutions for that skill or activity existing around you. The furthest such solutions from you define the edge of your abilities, the edge of your possible successes. It's silly and reductionist but still useful, I think.
So there you are, surrounded by all your possible successful outcomes. The distance function here is the relative difficulty of achieving any given solution, with the easier ones closer and the harder ones farther away. In the same way that the speed of light limits how much of the universe is the observable universe, you're going to be limited in your understanding of your limits based on how hard you try. There you sit, wondering if you can write a novel, and if you don't try really hard you'll never know. If you're fearful that you can't and so don't try, then that becomes your limitation until you're ready to try, and then your observable limit is pushed out the harder you try.
Okay but what if you're not sitting there wondering if you can write a novel but instead you're trying to provide for your family? What if your hypothetical maximal solution space is constrained by other factors that change the distance function? Like, what if the society you call home makes it harder for people like you, statistically, to succeed? You could imagine everyone in the society with their solution spaces, with all different distance functions, where success for any given goal could look equally easy or difficult but in fact it isn't because of these constraining factors.
Brevity is the soul of wit I guess, and I really like how this cartoon speaks to the differences in folks' lived experiences.
It can be a challenge to really understand how things affect the ease with which we live our lives. One possible corollary to that quote above is: "The limits you surpass don't tell you much about the limits others face." It's easy to extrapolate from our individual experiences to some supposed understanding of the objective natural world, and without any consideration for the pitfalls of extrapolation we can develop a misguided sense of how lives "should" be lived.
Two concepts that derive in some way from this are the Dunning-Krueger effect and the popular Occam's Razor. The Dunning-Krueger effect is basically the tendency for people who don't know any better to think they do. This explains all the Facebook epidemiologists who think the mRNA COVID vaccines will alter your genetic code and turn you into a gay. Or all the social scientists on Twitter who deny the epidemic of police brutality against marginalized communities and talk about "black-on-black crime". They don't know any better, meaning they don't really have a good sense of just how much work goes into understanding the social and political dynamics to see the patterns, but critically they think they know better. There are also people with legitimate claims to expertise in one domain who believe that expertise transfers to other, especially unrelated domains. Jordan Peterson is a purportedly successful and brilliant psychological researcher and theorist, but he has made his fortune convincing people that feminism is destroying Western civilization and that young men need to clean their rooms as a means to achieving success in life.
In its generic interpretation, Occam's Razor is an adage that says, "The simplest solution is often the best one." Something closer to what he actually said was, "Entities should not be multiplied beyond necessity," which is to say that we shouldn't include more factors in our hypotheses than we need to explain a phenomenon. We could visualize this as starting from the center of a solution space, and trying all the paths closest to us, then venturing further out, and so on. We're usually not searching for an absolute truth, but an explanation that's "good enough". One problem with "good enough" is that it's either closely related to that absolute truth, like in the same direction but not as far, or it's based on a different metric. That means that our being able to solve a problem will still be limited by the tractability of some absolute solution, or limited by convenience or another unrelated constraint.
Most of the time I see Occam's Razor invoked it's kind of a variation on Dunning-Krueger: someone thinks they know enough about economics, for instance, to say that all that falderal about systemic oppression and its effect on prosperity doesn't really explain why some people succeed and others don't. In fact, they think, it's just that some people are lazy and others have grit. In their minds, that's a far simpler--and so superior--solution, despite not having enough expertise to have an informed opinion.
I don't have a thesis to summarize. I don't have the acumen, experience, or expertise to contribute anything of substance to fixing anything. But I do like the shape of things as seen from different angles, and I think there are surprising ways to twist our understanding of our separate and shared conditions, of our creative and ethical lives, that are worth sharing. The metric space, goofy as it is, is something that works for me, so I wanted to share.