A distance seems like one of the simplest things in the world. Take two points, measure the space between them, and you are done.
Except that you are not.
In mathematics, a distance is not necessarily something measured with a ruler. It is a rule that tells us how different, separated, or dissimilar two objects are. The objects might be points in space, but they could just as easily be words, probability distributions, rankings, shapes, networks, or sequences.
And there is rarely only one sensible way to measure the distance between them.
Consider two points on a map. Euclidean distance gives us the length of the straight line connecting them. Manhattan distance imagines that movement is restricted to horizontal and vertical streets. The two measures answer slightly different questions about the same pair of points.
Neither is inherently the “real” distance. Each depends on what kind of movement, structure, or difference we care about.
That idea has always fascinated me.
Years ago, browsing the Encyclopedia of Distances, I discovered an extraordinary catalogue of ways in which mathematicians have tried to formalize separation. Some were familiar, others wonderfully obscure, and some seemed almost comical when encountered for the first time. There are distances for objects for which the everyday word distance would hardly occur to us at all.
But the more examples one sees, the stranger the underlying concept becomes.
What exactly are we doing when we define a distance?
Distance as a choice
Suppose two observations differ on ten variables. How far apart are they?
Before answering, we have already made several decisions. Do all ten variables matter equally? Should large differences dominate small ones? Should variables be standardized? Is a difference of two units on one variable comparable to two units on another? What if some variables are numerical and others categorical?
The distance does not merely discover separation.
It defines what separation means.
This is obvious in data analysis, where changing the distance measure can change neighbourhoods, clusters, maps, and ultimately our interpretation of the data. But the same idea becomes more interesting when taken outside mathematics.
Two cities may be fifty kilometres apart and still feel very distant because there is no convenient road between them.
Two people may live thousands of kilometres apart and speak every day.
Two people may share a home and hardly speak at all.
Which pair is closer?
The question sounds metaphorical, but perhaps the metaphor is doing something mathematically familiar. We have changed the dimensions on which distance is being evaluated.
Physical proximity is only one coordinate.
The distance between two people
Imagine, for a moment, trying to construct a distance between two people.
What variables would we need?
Shared experiences. Trust. Time spent together. Familiarity. Affection. Resentment. Mutual understanding. Expectations. Memories.
Would these quantities receive equal weights?
Almost certainly not.
And would the same weighting system work for every pair of people?
Probably not either.
More strangely, our hypothetical distance might not even satisfy the properties mathematicians normally demand of a metric. The distance from A to B might not feel identical to the distance from B to A. Being close to one person might make us feel farther from another. The shortest route between two people might pass through a shared friend, a place, a memory, or an event.
Human distance may be badly behaved mathematics.
But that does not make the idea useless.
On the contrary, thinking this way makes something visible: closeness is always closeness with respect to something.
Distance and representation
There is another complication.
We rarely observe things exactly as they are. We observe representations of them.
A dataset represents people, objects, events, or processes through a finite set of variables. Once we calculate distances between rows of that dataset, we are not measuring the distance between the underlying people or objects themselves. We are measuring the distance between their representations according to the information we happened to record.
This distinction is easy to forget.
A statistical analysis can tell us that two observations are close. But close in what space?
Change the variables, the coding, the scaling, or the distance function, and the geometry changes.
Perhaps something similar happens outside statistics. The distance we perceive between ourselves and another person depends partly on the representation we have constructed of that person — and of ourselves.
Memory changes the coordinates.
So does time.
Distances everywhere
Once you start thinking about distance this way, it becomes difficult to stop.
What is the distance between a memory and what actually happened?
Between an intention and an action?
Between what a scientific model measures and what we think it measures?
Between two political positions?
Between certainty and truth?
Between the person we are and the person we once expected to become?
Some of these questions may admit useful formalizations. Others probably should not. I am not interested in forcing every human experience into an equation.
What interests me is the underlying habit of thought.
To speak about distance is to ask what dimensions matter, how differences should be weighted, what counts as similarity, and what kind of path connects one thing to another.
That is already a surprisingly rich way of looking at the world.
Notes on Distance
This is what I want these notes to explore.
Sometimes the subject will be mathematical: unusual distances, clustering, representations, similarities and dissimilarities.
Sometimes it may be statistical.
Sometimes it may have almost nothing to do with statistics.
I have even had, for some time, the somewhat eccentric idea of writing a collection of stories built around different kinds of distance — the distance within a couple, between generations, between places, between memory and reality, perhaps even between people who believe themselves to be close.
For now, these notes are a place to collect such thoughts.
Because perhaps the most interesting question is not:
How far apart are these two things?
but rather:
What would it mean for them to be close?